[seqfan] Re: Canonical bijection from positive integers to positive rationals. What about zeros*?
I.V. Serov
i.v.serov at chf.nu
Thu Mar 5 23:00:05 CET 2020
Dear SeqFans, OEIS!
Please have a look at the following:
The Stern-Brocot-Tree, The-Tree-of-Knowledge:
https://oeis.org/A287732/a287732_1.png
as A287732 / A2887731 containing all rationals <-> all naturals.
The Pythagoras-Gauss-Tree, The-Tree-of-Life:
https://oeis.org/A287824/a287824_1.png
as A287824 / A287823 containing all fractions*, reduced and unreduced <-> odd
and even natural numbers.
Sincerely yours,
I. V. Serov
> On March 5, 2020 at 7:20 PM seqfan-request at list.seqfan.eu wrote:
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> Today's Topics:
>
> 1. Canonical bijection from positive integers to positive
> rationals. (Frank Adams-watters)
> 2. Re: Canonical bijection from positive integers to positive
> rationals. (Neil Sloane)
> 3. Re: Canonical bijection from positive integers to positive
> rationals. (Peter Luschny)
> 4. Fwd: Canonical bijection from positive integers to positive
> rationals. (Frank Adams-watters)
> 5. Re: Canonical bijection from positive integers to positive
> rationals. (Peter Munn)
> 6. Primes describing digit positions (?ric Angelini)
> 7. Re: Canonical bijection from positive integers to positive
> rationals. (Peter Munn)
> 8. Re: Primes describing digit positions -- follow up (?ric Angelini)
> 9. Re: Primes describing digit positions (Hans Havermann)
> 10. Re: Canonical bijection from positive integers to positive
> rationals. (Neil Sloane)
> 11. Re: I need help with defining these 3 sequences (M. F. Hasler)
> 12. Re: I need help with defining these 3 sequences (Ali Sada)
> 13. Re: Canonical bijection from positive integers to positive
> rationals. (Frank Adams-watters)
> 14. Re: Canonical bijection from positive integers to positive
> rationals. (Antti Karttunen)
> 15. Re: Canonical bijection from positive integers to positive
> rationals. (Frank Adams-watters)
> 16. Re: Canonical bijection from positive integers to positive
> rationals. (Antti Karttunen)
> 17. Re: Canonical bijection from positive integers to positive
> rationals. (Peter Munn)
> 18. Re: I need help with defining these 3 sequences (M. F. Hasler)
> 19. Re: Canonical bijection from positive integers to positive
> rationals. (Neil Sloane)
> 20. Re: Fwd: Canonical bijection from positive integers to
> positive rationals. (Peter Munn)
> 21. A sequence that moves itself (?ric Angelini)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Mon, 2 Mar 2020 15:49:24 +0000 (UTC)
> From: Frank Adams-watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Canonical bijection from positive integers to
> positive rationals.
> Message-ID: <213235072.2165346.1583164164842 at mail.yahoo.com>
> Content-Type: text/plain; charset=UTF-8
>
> There are two pairs of sequences claiming to be the canonical bijection from
> positive integers to positive rationals:
>
> A020652/A020653, and
> A038568/A038569.
>
> Which do we want to actually be our canonical sequence?
>
> I favor A020652/A020653. It seems cleaner to me.
>
> Compare A038566/A038567, which is the basis for both of these.
>
> Franklin T. Adams-Watters
>
>
> ------------------------------
>
> Message: 2
> Date: Mon, 2 Mar 2020 11:28:06 -0500
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAAOnSgTLh=i7GNUDquQC0-jT_DKYGX5uVEyMsa0T9tUKa72duA at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> I don't think we need to choose an official OEIS map, that would be
> presumptuous. And would not have much effect.
>
> For me, without doubt, the best map is the classic map based on the
> Stern's diatomic series (or Stern-Brocot sequence, from 1858: A002487.
> Look at all the references there.
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Mon, Mar 2, 2020 at 11:14 AM Frank Adams-watters via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
> > There are two pairs of sequences claiming to be the canonical bijection
> > from positive integers to positive rationals:
> >
> > A020652/A020653, and
> > A038568/A038569.
> >
> > Which do we want to actually be our canonical sequence?
> >
> > I favor A020652/A020653. It seems cleaner to me.
> >
> > Compare A038566/A038567, which is the basis for both of these.
> >
> > Franklin T. Adams-Watters
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
> ------------------------------
>
> Message: 3
> Date: Mon, 2 Mar 2020 17:58:52 +0100
> From: Peter Luschny <peter.luschny at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAMMbGZaPTxVgr+wKb7gsGNQj0grdrei3b+8Xt7qpEV+9jFEcSw at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> NS> For me, without doubt, the best map is the classic map based
> NS> on the Stern's diatomic series (or Stern-Brocot sequence, A002487.
>
> For me it is the Euclid tree: A295515.
>
> NS> from 1858: Look at all the references there.
>
> Let's wait and see. In 160 years, we can count references again.
> If you have Maple you can also try this delightful implementation:
>
> magic := x -> 1/(1 + floor(x) - frac(x)):
>
> And then run:
>
> 0; do magic(%) od;
>
> Cheers, Peter
>
>
> ------------------------------
>
> Message: 4
> Date: Mon, 2 Mar 2020 17:18:35 +0000 (UTC)
> From: Frank Adams-watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Fwd: Canonical bijection from positive integers to
> positive rationals.
> Message-ID: <396889195.2225859.1583169515422 at mail.yahoo.com>
> Content-Type: text/plain; charset=UTF-8
>
>
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Frank Adams-watters <franktaw at netscape.net>
> To: njasloane <njasloane at gmail.com>
> Sent: Mon, Mar 2, 2020 11:17 am
> Subject: Re: [seqfan] Canonical bijection from positive integers to positive
> rationals.
>
> Let me back off a minute here. I am planning on submitting a number of
> sequences encapsulating functions on positive rationals. Do we have a standard
> way to do this? If not, what should the standard be?
>
> I would think that one needs to choose a bijection s: Z+ -> Q+, and then if
> the function is f() the sequence is a(n) = f(s(n)). It was my understanding
> that the "canonical" bijection was precisely the s() to use for this. If not,
> then I don't understand what "canonical" means in a sequence.
>
> I am familiar with Stern-Brocot, and I agree that it is very beautiful. I
> didn't include it in the original message because it doesn't claim to be
> canonical. If we wind up using it for this purpose, I will have no objection.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Cc: Frank Adams-watters <franktaw at netscape.net>
> Sent: Mon, Mar 2, 2020 10:28 am
> Subject: Re: [seqfan] Canonical bijection from positive integers to positive
> rationals.
>
> I don't think we need to choose an official OEIS map, that would be
> presumptuous.? And would not have much effect.
> For me, without doubt, ?the best map is the classic map based on the Stern's
> diatomic series (or Stern-Brocot sequence, ?from 1858: A002487.? Look at all
> the references there.
> Best regardsNeil?
> Neil J. A. Sloane, President, OEIS Foundation.11 South Adelaide Avenue,
> Highland Park, NJ 08904, USA.Also Visiting Scientist, Math. Dept., Rutgers
> University, Piscataway, NJ.Phone: 732 828 6098; home page:
> http://NeilSloane.comEmail: njasloane at gmail.com
>
>
> On Mon, Mar 2, 2020 at 11:14 AM Frank Adams-watters via SeqFan
> <seqfan at list.seqfan.eu> wrote:
> There are two pairs of sequences claiming to be the canonical bijection from
> positive integers to positive rationals:
>
> A020652/A020653, and
> A038568/A038569.
>
> Which do we want to actually be our canonical sequence?
>
> I favor A020652/A020653. It seems cleaner to me.
>
> Compare A038566/A038567, which is the basis for both of these.
>
> Franklin T. Adams-Watters
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
> ------------------------------
>
> Message: 5
> Date: Mon, 2 Mar 2020 20:08:17 -0000
> From: "Peter Munn" <techsubs at pearceneptune.co.uk>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <d5f5a92efe8f240541c4d3ff598b357e.squirrel at webmail.nsnoc.com>
> Content-Type: text/plain;charset=iso-8859-1
>
> On Mon, March 2, 2020 4:58 pm, Peter Luschny wrote:
> > NS> For me, without doubt, the best map is the classic map based
> > NS> on the Stern's diatomic series (or Stern-Brocot sequence, A002487.
> >
> > For me it is the Euclid tree: A295515.
>
> I cast my vote for the Sagher map, A071974(n)/A071975(n), which is nicely
> multiplicative.
>
> Best Regards,
>
> Peter
>
> > NS> from 1858: Look at all the references there.
> >
> > Let's wait and see. In 160 years, we can count references again.
> > If you have Maple you can also try this delightful implementation:
> >
> > magic := x -> 1/(1 + floor(x) - frac(x)):
> >
> > And then run:
> >
> > 0; do magic(%) od;
> >
> > Cheers, Peter
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
>
>
> ------------------------------
>
> Message: 6
> Date: Mon, 2 Mar 2020 18:52:21 +0100 (CET)
> From: ?ric Angelini <bk263401 at skynet.be>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Primes describing digit positions
> Message-ID:
> <1326038166.10934.1583171541467 at webmail.appsuite.proximus.be>
> Content-Type: text/plain; charset=UTF-8
>
> Hello SeqFans,
> S = 11, 41, 61, 83, 113, 101, ...
> 11 says: "In position 1, there is 1";
> 41 says: "In position 4, there is 1";
> 61 says: "In position 6, there is 1";
> 83 says: "In position 8, there is 3";
> 113 says: "In position 11, there is 3";
> 101 says: "In position 10, there is 1";
> etc.
> We want the lexico-first seq of distinct
> positive primes "saying" the truth, like
> above, of course.
>
> But here is the tricky sequence T:
> "Lexico-first seq of distinct positive
> primes describing the position of every
> prime digit in T".
>
> What could be a(1) in T?
>
> Well a(1) cannot start with a digit 1, as
> a(1) = 11 (though a prime term "saying
> the truth") describes the position of 1
> -- and this 1 is not a prime digit;
> a(1) = 23 describes the position of a prime
> digit, yes, (3) but the digit "2" cannot
> be used in T! Indeed, as 2 is a prime digit,
> any description of it's position in T will
> end in "2" -- leadind to a term that is
> not prime!
> a(1) = 30 is wrong (not a prime term);
> a(1) = 31 is wrong too as "31" is not the
> description of a prime digit;
> a(1) = 32 is wrong (due to the "2")
> a(1) = 33 --> 36 are not primes;
> a(1) = 37 might be ok -- but as we have
> a "7" in second position, we will write
> "27" at some point in T -- with a forbidden
> "2"!
> So, a(1) starts perhaps with "4" and the
> digit "a" (below) cannot be prime;
>
> T = 4a, . . .
>
> This leaves us with a(1) = 41 only (because
> 41 is the only available 2-digit prime); is
> a(2) a 2-digit term? Let's see:
>
> T = 41, b1, . . (dots are empty positions)
>
> "b" cannot be 3 (as b is in 3rd position,
> this will lead in the future to "33", which
> is the description of b's position and a
> composite term);
> "b" cannot be 4 (41 is already in T);
> "b" cannot be 5 (51 is not a prime term);
> "b" cannot be 6 (as b is in 3rd position,
> this will lead in the future to "36", which
> is the description of b's position and a
> composite term);
> "b" might be "7" (no apparent contradiction
> yet, it seems):
>
> T = 41, 71, . . 1 . .
>
> Is a(3) a 2-digit term? Let's see:
>
> T = 41, 71, cd, 1 . .
>
> [this supposes that a(4) is at least a 3-digit
> term, as one can see]; we try "37" for a(3) as
> this term describes the position of "7" in T:
>
> T = 41, 71, 37, 1 . .
>
> This "37" will produce in the future the terms
> 53 and 67 (the description of "37"), which are
> luckily two primes); let's go on filling the
> positions 8 = e and 9 = f:
>
> T = 41, 71, 37, 1ef, . . .
>
> What if we use 0 for e and 1 for f? a(4) is a
> prime term at least:
>
> T = 41, 71, 37, 101, 1 . .
>
> We see that a(5) is a 3-digit term at least--
> we must thus wait in using the terms "53" and
> "67" we've seen above.
> Etc.
>
> Best (modulo errors)
> ?.
>
>
> ------------------------------
>
> Message: 7
> Date: Tue, 3 Mar 2020 09:23:49 -0000
> From: "Peter Munn" <techsubs at pearceneptune.co.uk>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <f08c717951d26592c93a7e593a40ab88.squirrel at webmail.nsnoc.com>
> Content-Type: text/plain;charset=iso-8859-1
>
> Update: Having found another nice bijection in A229994(n)/A077610(n), I
> have listed this and others mentioned in this thread in the crossrefs for
> A071974.
>
> If seqfans add any other bijective mappings that are in OEIS, then at some
> point these can be copied to the other sequences.
>
> Best Regards,
> Peter
>
> On Mon, March 2, 2020 8:08 pm, Peter Munn wrote:
> > On Mon, March 2, 2020 4:58 pm, Peter Luschny wrote:
> >> NS> For me, without doubt, the best map is the classic map based NS> on
> the Stern's diatomic series (or Stern-Brocot sequence, A002487. For me
> it is the Euclid tree: A295515.
> > I cast my vote for the Sagher map, A071974(n)/A071975(n), which is
> nicely
> > multiplicative.
> > Best Regards,
> > Peter
> >> NS> from 1858: Look at all the references there.
> >> Let's wait and see. In 160 years, we can count references again. If you
> have Maple you can also try this delightful implementation:
> >> magic := x -> 1/(1 + floor(x) - frac(x)):
> >> And then run:
> >> 0; do magic(%) od;
> >> Cheers, Peter
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
>
>
>
> ------------------------------
>
> Message: 8
> Date: Tue, 3 Mar 2020 12:43:52 +0100
> From: ?ric Angelini <eric.angelini at skynet.be>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Primes describing digit positions -- follow up
> Message-ID: <95196A31-68E0-42A4-9873-A28396E1B957 at skynet.be>
> Content-Type: text/plain; charset=utf-8
>
>
> I wrote (too quickly):
>
> > we must thus wait in using the terms "53" and "67" we've seen above.
>
> ... NO! The same argument used to
> forbid the digit 2 in T applies for the
> digit 5, of course.
> Mmmmmh, this leaves very few
> possibilities for T to exist.
> ?+
> ?.
> Catapult? de mon aPhone
>
>
> > Le 2 mars 2020 ? 18:52, ?ric Angelini <bk263401 at skynet.be> a ?crit :
> >
> > Hello SeqFans,
> > S = 11, 41, 61, 83, 113, 101, ...
> > 11 says: "In position 1, there is 1";
> > 41 says: "In position 4, there is 1";
> > 61 says: "In position 6, there is 1";
> > 83 says: "In position 8, there is 3";
> > 113 says: "In position 11, there is 3";
> > 101 says: "In position 10, there is 1";
> > etc.
> > We want the lexico-first seq of distinct
> > positive primes "saying" the truth, like
> > above, of course.
> >
> > But here is the tricky sequence T:
> > "Lexico-first seq of distinct positive
> > primes describing the position of every
> > prime digit in T".
> >
> > What could be a(1) in T?
> >
> > Well a(1) cannot start with a digit 1, as
> > a(1) = 11 (though a prime term "saying
> > the truth") describes the position of 1
> > -- and this 1 is not a prime digit;
> > a(1) = 23 describes the position of a prime
> > digit, yes, (3) but the digit "2" cannot
> > be used in T! Indeed, as 2 is a prime digit,
> > any description of it's position in T will
> > end in "2" -- leadind to a term that is
> > not prime!
> > a(1) = 30 is wrong (not a prime term);
> > a(1) = 31 is wrong too as "31" is not the
> > description of a prime digit;
> > a(1) = 32 is wrong (due to the "2")
> > a(1) = 33 --> 36 are not primes;
> > a(1) = 37 might be ok -- but as we have
> > a "7" in second position, we will write
> > "27" at some point in T -- with a forbidden
> > "2"!
> > So, a(1) starts perhaps with "4" and the
> > digit "a" (below) cannot be prime;
> >
> > T = 4a, . . .
> >
> > This leaves us with a(1) = 41 only (because
> > 41 is the only available 2-digit prime); is
> > a(2) a 2-digit term? Let's see:
> >
> > T = 41, b1, . . (dots are empty positions)
> >
> > "b" cannot be 3 (as b is in 3rd position,
> > this will lead in the future to "33", which
> > is the description of b's position and a
> > composite term);
> > "b" cannot be 4 (41 is already in T);
> > "b" cannot be 5 (51 is not a prime term);
> > "b" cannot be 6 (as b is in 3rd position,
> > this will lead in the future to "36", which
> > is the description of b's position and a
> > composite term);
> > "b" might be "7" (no apparent contradiction
> > yet, it seems):
> >
> > T = 41, 71, . . 1 . .
> >
> > Is a(3) a 2-digit term? Let's see:
> >
> > T = 41, 71, cd, 1 . .
> >
> > [this supposes that a(4) is at least a 3-digit
> > term, as one can see]; we try "37" for a(3) as
> > this term describes the position of "7" in T:
> >
> > T = 41, 71, 37, 1 . .
> >
> > This "37" will produce in the future the terms
> > 53 and 67 (the description of "37"), which are
> > luckily two primes); let's go on filling the
> > positions 8 = e and 9 = f:
> >
> > T = 41, 71, 37, 1ef, . . .
> >
> > What if we use 0 for e and 1 for f? a(4) is a
> > prime term at least:
> >
> > T = 41, 71, 37, 101, 1 . .
> >
> > We see that a(5) is a 3-digit term at least--
> > we must thus wait in using the terms "53" and
> > "67" we've seen above.
> > Etc.
> >
> > Best (modulo errors)
> > ?.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> ------------------------------
>
> Message: 9
> Date: Tue, 3 Mar 2020 11:11:13 -0500
> From: Hans Havermann <gladhobo at bell.net>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Primes describing digit positions
> Message-ID: <9C0E09D9-B119-462F-B9F0-FD8B9D99E448 at bell.net>
> Content-Type: text/plain; charset=us-ascii
>
> EA: "S = 11, 41, 61, 83, 113, 101, ... But here is the tricky sequence T"
>
> S is tricky enough! By hand, I find ..., 541, 1009, 599, ..., 991, 9001, 1031,
> ...
>
>
>
> ------------------------------
>
> Message: 10
> Date: Tue, 3 Mar 2020 13:05:22 -0500
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAAOnSgS7=Xu2G=9BxfyQKxf+hbNGAsY2pri7iwgswYdPKTewtA at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> There is an entry in the Index to the OEIS, Section Ra, under "rational
> numbers" which lists several of these bijections.
> Peter or Frank, could you please expand this Index entry with any bijective
> mappings that are not there.
>
> And also add a link to the index entry to all of these sequence.
> The link should say:
>
> <a href="/index/Ra#rational">Index entries for sequences related to
> enumerating the rationals</a>
>
>
> By the way, there are also cross-references to this Index entry from
> entries in the Index for "Bijection ..." and "Enumerating ..."
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Tue, Mar 3, 2020 at 10:17 AM Peter Munn <techsubs at pearceneptune.co.uk>
> wrote:
>
> > Update: Having found another nice bijection in A229994(n)/A077610(n), I
> > have listed this and others mentioned in this thread in the crossrefs for
> > A071974.
> >
> > If seqfans add any other bijective mappings that are in OEIS, then at some
> > point these can be copied to the other sequences.
> >
> > Best Regards,
> > Peter
> >
> > On Mon, March 2, 2020 8:08 pm, Peter Munn wrote:
> > > On Mon, March 2, 2020 4:58 pm, Peter Luschny wrote:
> > >> NS> For me, without doubt, the best map is the classic map based NS> on
> > the Stern's diatomic series (or Stern-Brocot sequence, A002487. For me
> > it is the Euclid tree: A295515.
> > > I cast my vote for the Sagher map, A071974(n)/A071975(n), which is
> > nicely
> > > multiplicative.
> > > Best Regards,
> > > Peter
> > >> NS> from 1858: Look at all the references there.
> > >> Let's wait and see. In 160 years, we can count references again. If you
> > have Maple you can also try this delightful implementation:
> > >> magic := x -> 1/(1 + floor(x) - frac(x)):
> > >> And then run:
> > >> 0; do magic(%) od;
> > >> Cheers, Peter
> > >> --
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> >
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
> ------------------------------
>
> Message: 11
> Date: Tue, 3 Mar 2020 18:55:37 -0400
> From: "M. F. Hasler" <seqfan at hasler.fr>
> To: Ali Sada <pemd70 at yahoo.com>
> Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: I need help with defining these 3 sequences
> Message-ID:
> <CABxCbJ16bbVRcwkgy=2v+enxMWG0Gb1vLSy+5qFnszKRMC-3hw at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> On Sun, Mar 1, 2020 at 2:30 AM Ali Sada wrote:
>
> > Sequence 2:
> > We start with a(1) = 1 and a(2) = 2.
> > a(3) = a(1) + a(2) = 3. We don?t have any gap here, so we continue.
> > We add a(2) and a(3) and we get 5. So, a(5) = 5. To find a(4), we add a(5)
> > + a(3) = 8.
> > We filled the gap between 3 and 5, so we continue.
> > We add a(4) + a(5) = 5+8= 13, so a(13) =1 3. Now, we need to fill the gaps
> > between 6 and 12.
> > a(6) = a(5) + a(13) = 5+13 = 18
> > a(12) = a(6) + a(13) = (...)
> >
>
> This is a variant of the Fibonacci numbers F(n) = F(n-1)+F(n-2).
> While there is a "gap" to fill [say, between a(n) and a(n+m)], you place
> alternatively such sums to the left and to the right of the gap.
> then you continue with a(X) = X = a(n+m) + a(n+m-1) which is changed w.r.t.
> Fibonacci formula because here
> a(n+m-1) = a(n+m) + a(n+1) = 2 a(n+m) + a(n).
>
> Maybe I did n't understand correctly, but I think you made an error in the
> 2nd-to-last line above (so all subsequent terms are wrong):
> I think it should be
> a(6) = a(5) + a(13) = ***8*** + 13 = F(6) + F(7) = F(8) = 21
> (is this correct?)
> and then
> a(12) = a(6) + a(13) = 21 + 13 = F(8) + F(7) = F(9) = 34
> (followed by a(7) = a(6)+a(12) = F(8)+F(9) = F(10), a(11) = a(7)+a(12) =
> F(10)+F(9) = F(11), etc)
> and later
> a(13) + a(12) = F(7) + F(9) = 13 + 34 = 47 =: a(47)
> I think this is the first term which is not a Fibonacci number
> (because of the gap being larger than 1 as for (a(3), a(5)), and therefore
> it has F(k+2) as last term and not just F(k+1)).
>
> If we used, to continue once a gap is filled, a(n+m) and a(n+1) instead of
> a(n+m-1), then the sequence would consist only of Fibonacci numbers:
> a(13)+a(6) = F(7)+ F(8) = F(9) = 34 =: a(34)
> a(34)+a(14) = F(9)+ F(10) = F(11) = 89 =: a(89)
> a(89)+a(35) = F(11)+ F(12) = F(13) = 233 =: a(233) etc.
>
> - Maximilian
>
>
> ------------------------------
>
> Message: 12
> Date: Wed, 4 Mar 2020 00:40:05 +0000 (UTC)
> From: Ali Sada <pemd70 at yahoo.com>
> To: "M. F. Hasler" <seqfan at hasler.fr>
> Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: I need help with defining these 3 sequences
> Message-ID: <358832318.3968494.1583282405850 at mail.yahoo.com>
> Content-Type: text/plain; charset=UTF-8
>
> Thank you very much Dr. Hasler. I really appreciate your response.
> Again, my language didn't help me convey what I wanted to say.
> After we fill the first five terms, we add a(4) + a(5) = 13. So, a(13)
> =13.Now, we have 7 gaps between a(5) and a(13).Since a(13) is on the right, we
> fill the first gap on the left: a(6) = a(13) + a(5) = 18.Then, we fill the
> first gap on the right: a(12)= a(6) + a(13) = 18+13 = 31.We fill the first gap
> on the left: a(7) = a(6) + a(12) = 18+31 = 49.We fill the first gap on the
> right: a(11) = a(7) + a(12) = 49+ 31 = 80.We fill the first gap on the left:
> a(8) = a(7) + a(11) = 49+80 =129.We fill the first gap on the right: a(10)=
> a(8) + a(11) = 129+80 = 209.We fill the first gap on the? left: a(9) = a(8) +
> a(10) = 129+209 = 338.
> Now we don't have any more gaps, so we continue. We add a(12) + a(13) = 13+31=
> 44. So, a(44) = 44, and we repeat the same procedure above.
> It is much easier to show it on a blackboard! Please see this image.
> https://justpaste.it/3pflgIn each row, we add the two green numbers to get the
> yellow number.
> Thank you again for your help.
> Best,
> Ali
>
>
>
>
>
>
>
>
> On Tuesday, March 3, 2020, 5:56:08 PM EST, M. F. Hasler <seqfan at hasler.fr>
> wrote:
>
> On Sun, Mar 1, 2020 at 2:30 AM Ali Sada wrote:
>
> Sequence 2:?
> We start with a(1) = 1 and a(2) = 2.
> a(3) = a(1) + a(2) = 3. We don?t have any gap here, so we continue.
> We add a(2) and a(3) and we get 5. So, a(5) = 5. To find a(4), we add a(5) +
> a(3) = 8.
> We filled the gap between 3 and 5, so we continue.
> We add a(4) + a(5) = 5+8= 13, so a(13) =1 3. Now, we need to fill the gaps
> between 6 and 12.
> a(6) = a(5) + a(13) = 5+13 = 18
> a(12) = a(6) + a(13) = (...)
>
>
> This is a variant of the Fibonacci numbers F(n) = F(n-1)+F(n-2).While there is
> a "gap" to fill [say, between a(n) and a(n+m)], you place alternatively such
> sums to the left and to the right of the gap.then you continue with a(X) = X =
> a(n+m)?+ a(n+m-1) which is changed w.r.t. Fibonacci formula because here
> a(n+m-1) = a(n+m)?+ a(n+1) = 2 a(n+m)?+ a(n).?
> Maybe I did n't understand correctly, but I think you made an error in the
> 2nd-to-last line above (so all subsequent terms are wrong):I think it should
> bea(6) = a(5) + a(13) = ***8*** + 13 = F(6)+?F(7) = F(8) = 21?(is this
> correct?)and thena(12) = a(6) + a(13) = 21?+ 13 = F(8)?+ F(7) = F(9) =
> 34(followed by a(7) = a(6)+a(12) = F(8)+F(9) = F(10),?a(11) = a(7)+a(12) =
> F(10)+F(9) = F(11), etc)and latera(13)?+ a(12) = F(7)?+ F(9) = 13?+ 34 = 47?=:
> a(47)
> I think this is the first term which is not a Fibonacci number(because of the
> gap being larger than 1 as for (a(3), a(5)), and therefore it has F(k+2) as
> last term and not just F(k+1)).
> If we used, to continue once a gap is filled, a(n+m) and a(n+1) instead of
> a(n+m-1), then the sequence would consist only of Fibonacci numbers:a(13)+a(6)
> = F(7)+F(8) = F(9) = 34 =: a(34)a(34)+a(14) = F(9)+F(10) = F(11) = 89 =:
> a(89)a(89)+a(35) = F(11)+F(12) = F(13) = 233 =: a(233) etc.
> - Maximilian
>
>
> ------------------------------
>
> Message: 13
> Date: Tue, 3 Mar 2020 15:46:50 +0000 (UTC)
> From: Frank Adams-watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID: <1683874848.2581537.1583250410447 at mail.yahoo.com>
> Content-Type: text/plain; charset=UTF-8
>
> This ought to be in the index, I would think.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Peter Munn <techsubs at pearceneptune.co.uk>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Tue, Mar 3, 2020 9:17 am
> Subject: [seqfan] Re: Canonical bijection from positive integers to positive
> rationals.
>
> Update: Having found another nice bijection in A229994(n)/A077610(n), I
> have listed this and others mentioned in this thread in the crossrefs for
> A071974.
>
> If seqfans add any other bijective mappings that are in OEIS, then at some
> point these can be copied to the other sequences.
>
> Best Regards,
> Peter
>
> On Mon, March 2, 2020 8:08 pm, Peter Munn wrote:
> > On Mon, March 2, 2020 4:58 pm, Peter Luschny wrote:
> >> NS> For me, without doubt, the best map is the classic map based NS> on
> the Stern's diatomic series (or Stern-Brocot sequence, A002487. For me
> it is the Euclid tree: A295515.
> > I cast my vote for the Sagher map, A071974(n)/A071975(n), which is
> nicely
> > multiplicative.
> > Best Regards,
> >? Peter
> >> NS> from 1858: Look at all the references there.
> >> Let's wait and see. In 160 years, we can count references again. If you
> have Maple you can also try this delightful implementation:
> >>? ? magic := x -> 1/(1 + floor(x) - frac(x)):
> >> And then run:
> >>? ? 0; do magic(%) od;
> >> Cheers, Peter
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
> ------------------------------
>
> Message: 14
> Date: Tue, 3 Mar 2020 18:16:37 +0200
> From: Antti Karttunen <antti.karttunen at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAB+0_=kOOrJ1fTHTT_H568fU-O5e90k6jQXhb-nFEpndj4aE6A at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> On 3/3/20, Peter Munn <techsubs at pearceneptune.co.uk> wrote:
> > Update: Having found another nice bijection in A229994(n)/A077610(n), I
> > have listed this and others mentioned in this thread in the crossrefs for
> > A071974.
>
> Peter(s),
>
> please see https://oeis.org/wiki/Index_to_OEIS:_Section_Fo#fraction_trees
> and update it, if the pairs of sequences you have mentioned can be
> viewed as trees in any way (like A002487).
>
> Best regards,
>
> Antti
>
>
> >
> > If seqfans add any other bijective mappings that are in OEIS, then at some
> > point these can be copied to the other sequences.
> >
> > Best Regards,
> > Peter
> >
> > On Mon, March 2, 2020 8:08 pm, Peter Munn wrote:
> >> On Mon, March 2, 2020 4:58 pm, Peter Luschny wrote:
> >>> NS> For me, without doubt, the best map is the classic map based NS> on
> > the Stern's diatomic series (or Stern-Brocot sequence, A002487. For me
> > it is the Euclid tree: A295515.
> >> I cast my vote for the Sagher map, A071974(n)/A071975(n), which is
> > nicely
> >> multiplicative.
> >> Best Regards,
> >> Peter
> >>> NS> from 1858: Look at all the references there.
> >>> Let's wait and see. In 160 years, we can count references again. If you
> > have Maple you can also try this delightful implementation:
> >>> magic := x -> 1/(1 + floor(x) - frac(x)):
> >>> And then run:
> >>> 0; do magic(%) od;
> >>> Cheers, Peter
> >>> --
> >>> Seqfan Mailing list - http://list.seqfan.eu/
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> >
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
> ------------------------------
>
> Message: 15
> Date: Tue, 3 Mar 2020 18:13:56 +0000 (UTC)
> From: Frank Adams-watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID: <1738213076.2641761.1583259236999 at mail.yahoo.com>
> Content-Type: text/plain; charset=UTF-8
>
> I still haven't gotten any answers to my fundamental question: how do I enter
> a function defined on the positive rationals into the database? Should I just
> pick one of these bijections to use?
>
> Franklin T. Adams-Watters
>
>
>
>
> ------------------------------
>
> Message: 16
> Date: Wed, 4 Mar 2020 11:28:36 +0200
> From: Antti Karttunen <antti.karttunen at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Cc: Frank Adams-watters <franktaw at netscape.net>, Peter G Munn
> <pmunn at pearceneptune.co.uk>, Neil Sloane <njasloane at gmail.com>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAB+0_==A9eNyAHHGJSqCHea_f7qrqjqcdY3NXst8vQK3--biDA at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> On 3/3/20, Frank Adams-watters via SeqFan <seqfan at list.seqfan.eu> wrote:
> > I still haven't gotten any answers to my fundamental question: how do I
> > enter a function defined on the positive rationals into the database? Should
> > I just pick one of these bijections to use?
> >
>
> I would use A002487(n)/A002487(1+n) and maybe also Sagher-map
> A071974(n)/A071975(n) that was suggested by Peter, especially if the
> synergetic effect (with your chosen function) is something elegant
> (just my intuition, as both num & den of the latter are
> multiplicative).
>
> I guess the range of your functions is in Q? So one could store both
> the fraction's num and den (as is usual for fractions), but for any
> such fraction sequence, we could (also) "compress" that information to
> just the index n in the sequence of fractions like
> A002487(n)/A002487(1+n) or A071974(n)/A071975(n) (sometimes this could
> even reduce to something very simple?)
>
>
> Best,
>
> Antti
>
>
> > Franklin T. Adams-Watters
> >
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
> ------------------------------
>
> Message: 17
> Date: Wed, 4 Mar 2020 10:50:00 -0000
> From: "Peter Munn" <techsubs at pearceneptune.co.uk>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Cc: "Antti Karttunen" <antti.karttunen at gmail.com>, "Neil Sloane"
> <njasloane at gmail.com>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <b5220e8ad47e89fdb58df78e0d2ca6c2.squirrel at webmail.nsnoc.com>
> Content-Type: text/plain;charset=iso-8859-1
>
> Please excuse me. Three contributors to this thread have suggested I
> (and/or another) update the index, as though it is obvious how this is
> done. It is not.
>
> Antti, I can see you have made updates to the index. Could you edit the
> instruction page [1] to expand on the paragraph entitled "Adding a link
> from the Index to a sequence"?
>
> This paragraph consists only of the words: "Type {{A|123456}} to make a
> link to A123456 (say)." This described too little of the whole process
> that (the only time I tried) I got to a position where I wasn't sure I was
> doing it correctly, and even less sure of what to do next to complete the
> process. So rather than make a mess, I abandoned my attempt.
>
> (Yes, I know the page also advises, "If in doubt, consult one of the
> Editors-in-Chief". I did this, and it proved fruitless.)
>
> Thanks,
> Peter
>
> [1] https://oeis.org/wiki/Index:_Instructions_For_Updating_Index_to_OEIS
>
> On Tue, March 3, 2020 4:16 pm, Antti Karttunen wrote:
> > On 3/3/20, Peter Munn <techsubs at pearceneptune.co.uk> wrote:
> >> Update: Having found another nice bijection in A229994(n)/A077610(n), I
> >> have listed this and others mentioned in this thread in the crossrefs
> >> for
> >> A071974.
> >
> > Peter(s),
> >
> > please see https://oeis.org/wiki/Index_to_OEIS:_Section_Fo#fraction_trees
> > and update it, if the pairs of sequences you have mentioned can be
> > viewed as trees in any way (like A002487).
> >
> > Best regards,
> >
> > Antti
> >
> >
> >>
> >> If seqfans add any other bijective mappings that are in OEIS,
> >> then at some point these can be copied to the other sequences.
> >>
> >> Best Regards,
> >> Peter
> >>
>
>
>
> ------------------------------
>
> Message: 18
> Date: Wed, 4 Mar 2020 10:07:16 -0400
> From: "M. F. Hasler" <seqfan at hasler.fr>
> To: Ali Sada <pemd70 at yahoo.com>
> Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: I need help with defining these 3 sequences
> Message-ID:
> <CABxCbJ06w_jampJMi8ekEKrHa__Gf3mwUxe9NfpQE+11paW44w at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> On Tue, Mar 3, 2020 at 8:40 PM Ali Sada wrote:
>
> > After we fill the first five terms, we add a(4) + a(5) = 13. So, a(13) =13.
> > Now, we have 7 gaps between a(5) and a(13).
> > Since a(13) is on the right, we fill the first gap on the left: a(6) =
> > a(13) + a(5) = 18.
> >
>
> Oh yes, I was misled by:
> " We add a(4) + a(5) = 5+8 = ... " in the earlier mail.
> This made me think that a(5) = 8 instead of 5. Sorry for this mistake!
> I should have realized it since the borders of the intervals are always
> a(m) = m...
>
> So yes, indeed, the deviation from Fibonacci's occurs already with a(6) =
> 18.
>
> In (PARI):
> a=Vec(m=1,44); while(#a >= m = a[n=m]+a[n-(n>1)], a[m]=m; for(k=1,m-n-1, a[
> if( k%2, n+k\/2, m-k\2 )] = a[n+k\2]+a[m-k\/2+1] ));a
> = [1, 2, 3, 8, 5, 18, 49, 129, 338, 209, 80, 31, 13, 57, 158, 417, 1093,
> 2862, 7493, 19617, 51358, 134457, 352013, 921582, 2412733, 6316617,
> 16537118, 43294737, 70052356, 26757619, 10220501, 3903884, 1491151, 569569,
> 217556, 83099, 31741, 12124, 4631, 1769, 676, 259, 101, 44, ...]
>
> The sequence of "new upper ends" m' = a(m-1) + a(m) is
> [1, 2,] 3, 5, 13, 44, 145, 479, 1582, 5225, 17257, 56996, ...:
> As written earlier, after the gap (3,5) with only one "hole", we have
> a(m-1) = a(n+1) + a(m) = a(n) + a(m)*2
> (where n is the left end of the gap = predecessor of m), and thus m' =
> a(m-1)+a(m) = a(n) + a(m)*3,
> i.e., m(i) = 3 m(i-1) + m(i-2) after m(1..5) = (1, 2, 3, 5, 13).
>
> (PARI) for(n=1+#a=[1,2,3,5,13],#a=Vec(a,100), a[n]=a[n-1]*3+a[n-2]);a
>
> - Maximilian
>
>
> ------------------------------
>
> Message: 19
> Date: Wed, 4 Mar 2020 10:15:32 -0500
> From: Neil Sloane <njasloane at gmail.com>
> To: Peter Munn <techsubs at pearceneptune.co.uk>
> Cc: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>, Antti
> Karttunen <antti.karttunen at gmail.com>
> Subject: [seqfan] Re: Canonical bijection from positive integers to
> positive rationals.
> Message-ID:
> <CAAOnSgRHSb5S5ZeuDkC1-y4=eBE56+=jL6O+sm_jvENZN9Bctw at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> Peter Munn,
> Remember that to edit a page on the wiki you must first login to the wiki.
> (This is different from logging in to the database)
> Then you can go to a page - and Index page say - and click edit at the top
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Wed, Mar 4, 2020 at 5:50 AM Peter Munn <techsubs at pearceneptune.co.uk>
> wrote:
>
> > Please excuse me. Three contributors to this thread have suggested I
> > (and/or another) update the index, as though it is obvious how this is
> > done. It is not.
> >
> > Antti, I can see you have made updates to the index. Could you edit the
> > instruction page [1] to expand on the paragraph entitled "Adding a link
> > from the Index to a sequence"?
> >
> > This paragraph consists only of the words: "Type {{A|123456}} to make a
> > link to A123456 (say)." This described too little of the whole process
> > that (the only time I tried) I got to a position where I wasn't sure I was
> > doing it correctly, and even less sure of what to do next to complete the
> > process. So rather than make a mess, I abandoned my attempt.
> >
> > (Yes, I know the page also advises, "If in doubt, consult one of the
> > Editors-in-Chief". I did this, and it proved fruitless.)
> >
> > Thanks,
> > Peter
> >
> > [1] https://oeis.org/wiki/Index:_Instructions_For_Updating_Index_to_OEIS
> >
> > On Tue, March 3, 2020 4:16 pm, Antti Karttunen wrote:
> > > On 3/3/20, Peter Munn <techsubs at pearceneptune.co.uk> wrote:
> > >> Update: Having found another nice bijection in A229994(n)/A077610(n), I
> > >> have listed this and others mentioned in this thread in the crossrefs
> > >> for
> > >> A071974.
> > >
> > > Peter(s),
> > >
> > > please see
> > https://oeis.org/wiki/Index_to_OEIS:_Section_Fo#fraction_trees
> > > and update it, if the pairs of sequences you have mentioned can be
> > > viewed as trees in any way (like A002487).
> > >
> > > Best regards,
> > >
> > > Antti
> > >
> > >
> > >>
> > >> If seqfans add any other bijective mappings that are in OEIS,
> > >> then at some point these can be copied to the other sequences.
> > >>
> > >> Best Regards,
> > >> Peter
> > >>
> >
> >
>
>
> ------------------------------
>
> Message: 20
> Date: Wed, 4 Mar 2020 15:28:24 -0000
> From: "Peter Munn" <techsubs at pearceneptune.co.uk>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Fwd: Canonical bijection from positive integers
> to positive rationals.
> Message-ID:
> <93c3fff68bb1b0b0e8c076dc4f1ef9b7.squirrel at webmail.nsnoc.com>
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>
> On Mon, March 2, 2020 5:18 pm, Frank Adams-watters via SeqFan wrote:
> > From: Frank Adams-watters <franktaw at netscape.net>
> > To: njasloane <njasloane at gmail.com>
> > Sent: Mon, Mar 2, 2020 11:17 am
> >
> > Let me back off a minute here. I am planning on submitting a number of
> > sequences encapsulating functions on positive rationals. Do we have a
> > standard way to do this? If not, what should the standard be?
> >
> > I would think that one needs to choose a bijection s: Z+ -> Q+, and then
> > if the function is f() the sequence is a(n) = f(s(n)). It was my
> > understanding that the "canonical" bijection was precisely the s() to use
> > for this. If not, then I don't understand what "canonical" means in a
> > sequence.
>
> I can see that it would be useful for there not to be a lot of different
> mappings, s: Z+ -> Q+ used in OEIS for functions on rationals. For one
> thing, it would make it difficult to find them by searching for the
> sequence terms.
>
> So, I decided to see if I could find any of the following sequences in
> OEIS by looking for terms I had calculated:
> a(n) = s^-1(s(n) * 2)
> a(n) = s^-1(s(n) / 2)
> T(n,k) = s^-1(s(n) + s(k))
> T(n,k) = s^-1(s(n) * s(k)).
>
> I did this for s corresponding to A002487, A020652/A020653,
> A038568/A038569, A071974/A071975 and A229994/A077610.
>
> I drew a blank. (I gave up at that point, so I didn't try A295515 or any
> others. I had looked for both square and triangular forms of tables.)
>
> A different search technique could prove more fruitful, but unless anyone
> knows better, it may be you are heading into virgin territory, Franklin.
>
> FWIW, the mapping that seemed best at keeping the early terms small (in
> the above sequences) was A229994/A077610.
>
> Best Regards,
>
> Peter
>
> > I am familiar with Stern-Brocot, and I agree that it is very beautiful. I
> > didn't include it in the original message because it doesn't claim to be
> > canonical. If we wind up using it for this purpose, I will have no
> > objection.
> >
> > Franklin T. Adams-Watters
> >
> > -----Original Message-----
> > From: Neil Sloane <njasloane at gmail.com>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Cc: Frank Adams-watters <franktaw at netscape.net>
> > Sent: Mon, Mar 2, 2020 10:28 am
> > Subject: Re: [seqfan] Canonical bijection from positive integers to
> > positive rationals.
> >
> > I don't think we need to choose an official OEIS map, that would be
> > presumptuous.?? And would not have much effect.
> > For me, without doubt, ??the best map is the classic map based on the
> > Stern's diatomic series (or Stern-Brocot sequence, ??from 1858: A002487.??
> > Look at all the references there.
> > Best regardsNeil??
> > Neil J. A. Sloane, President, OEIS Foundation.
> >
> > On Mon, Mar 2, 2020 at 11:14 AM Frank Adams-watters via SeqFan
> > <seqfan at list.seqfan.eu> wrote:
> > There are two pairs of sequences claiming to be the canonical bijection
> > from positive integers to positive rationals:
> >
> > A020652/A020653, and
> > A038568/A038569.
> >
> > Which do we want to actually be our canonical sequence?
> >
> > I favor A020652/A020653. It seems cleaner to me.
> >
> > Compare A038566/A038567, which is the basis for both of these.
>
>
>
> ------------------------------
>
> Message: 21
> Date: Thu, 5 Mar 2020 20:09:09 +0100 (CET)
> From: ?ric Angelini <bk263401 at skynet.be>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] A sequence that moves itself
> Message-ID:
> <473896025.110823.1583435350001 at webmail.appsuite.proximus.be>
> Content-Type: text/plain; charset=UTF-8
>
> Hello SeqFans,
> Inspired by the wonderful work of Lars Blomberg, visible here
> https://oeis.org/A308387,
> we would like to develop more or less the same idea for the
> sequence M (M for Moving):
>
> a(n) will command to the a(n)th digit of M to jump over
> a(n) digits to the right and
> burry itself under the digit it lands on. When all jumps
> are done, the buried digits
> will reproduce the starting sequence -- commas included.
>
> (Yes, when the rightmost digit of a term jumps, it brings
> the comma with it in the grave).
>
> Now, what could be the lexico-1st such M of distinct positive
> terms? Well... I must admit I don't even know how to start M!
> (...)
> Follow up with example and color here, on my personal blog:
>
> https://bit.ly/2wsxpxf
>
> Best,
> ?.
>
>
> ------------------------------
>
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