[seqfan] INVERT-transforms and problems with Superseeker?
Antti Karttunen
Antti.Karttunen at iki.fi
Thu Dec 11 09:48:42 CET 2003
Thanks John, Thanks Christian,
Caution Neil!
First, an essential question:
Is there something wrong with the Superseeker?
That is, I cannot get same results from it, however I'll try
to lookup from different positions.
What is it exactly what superseeker does to the sequence
when it's looked up, and various transformations are applied
to it? E.g. are any terms from the front silently discarded
before the transformations? At least INVERT and its inverse
are very sensitive to the offsets.
It would help a lot if there would be a "debug-mode" (say lookupd)
which would give results of each hundred and more transformations
when applied to the looked up sequence, regardless of whether
they match against the database or not.
I will give some of the (longish) results by Superseeker in the end of
this mail,
but first my comments to John and Christian:
John wrote:
> Date: Mon, 8 Dec 2003 11:33:58 -0500 (EST)
> From: John Layman <layman at calvin.math.vt.edu>
> To: seqfan at ext.jussieu.fr
> Cc: John Layman <layman at calvin.math.vt.edu>
> Subject: Some INVERT transforms.
> Antti Karttunen (Dec 8, 2003): ... could you compute for me the
following transforms:
> INVERT([1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]);
> and also this:
>
INVERT([1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946]);
> Me: Here are some results that I obtained using my own ISAP (Integer
Sequence Analysis Program),
> written in Pascal (US denotes the entered USer sequence):
> US =1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (25)
> (INV)US = 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
6765 10946 17711 28657 46368 75025 121393 (25)
> Iterate Composite Transform (y/n)?
> US =1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (25)
> T=(INV); (T^2)US = 1 3 8 22 60 164 448 1224 3344 9136 24960 68192
186304 508992
> 1390592 3799168 10379520 28357376 77473792 211662336 578272256
1579869184
> 4316282880 11792304128 32217174016 (25)
> This indicates that (INV){1,1,0,0,0,0,0,...} is indeed the Fibonacci
seq. and that (INV)(INV){1,1,0,0,0,0,0,0,...}
> =(INV){1,2,3,5,8,13,21,34,55,...} is A028859
> John
Christian wrote:
> Date: Mon, 08 Dec 2003 11:07:11 -0800
> From: Christian G. Bower <bowerc at usa.net>
> To: seqfan at ext.jussieu.fr
> Subject: Re: [[seqfan] Few INVERT transforms requested. A000045 -->
A000129, how ?]
> Like John Laymen, I'm not using the Maple procedure, but rather my own
> program
>
INVERT([0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946]);
> I get:
> A000129 Pell Numbers
>
INVERT([1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946]);
> I get:
> A007484 E(2,7)
I.e.:
%S A007484 2,7,25,89,317,1129,4021,14321,51005,181657,646981,2304257,8206733,29228713, ...
quite different from A028859, which John got. But I guess
your programs interpret the offsets in different way,
with your program having the first term of the list for a(0),
while John's program interprets it as a(1) ?
And shouldn't the term a(0) be ignored in this transformation
in any case?
>
>> Should some of these result A000129?
>> And if you can prove it with g.f.ology or with some combinatorial
identity,
>> then even better.
> Don't have time to do proofs right now. Since Pell numbers are defined by
> simple recurrence, should be easy to convert to a g.f. and thus show
> equivalence to the transformations.
OK, this would be a good g.f.ological exercise for me, who is having
hard time in
reciprocating even a simple polynomial correctly! But maybe I will find
a combinatorial
proof instead, starting for example from that "left factors of Grand
Schroder Paths"
interpretation of A000129.
> Christian
BTW. Are either of the programs available for testing?
Yours,
Antti
PS. Here are some selected dumps from superseker:
==========================================================================================
For lookup 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
6765 10946
I got:
Transformation T109 gave a match with:
%I A085974
%S A085974 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,
%T A085974 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U A085974 0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0
%N A085974 Number of 0's in decimal expansion of prime(n).
%e A085974 prime(26) = 101, so a(26)=1 and prime(1230) = 10007, so
%e A085974 a(1230)=3.
%Y A085974 Cf. 1's A085975, 2's A085976, 3's A085977, 4's A085978, 5's A085979, 6's A085980, 7's A085981, 8's A085982, 9's A085983.
%Y A085974 Adjacent sequences: A085971 A085972 A085973 this_sequence A085975 A085976 A085977
%Y A085974 Sequence in context: A011742 A011741 A011740 this_sequence A011739 A023975 A011738
%K A085974 base,nonn
%O A085974 1,1
%A A085974 Jason Earls (jcearls(AT)kskc.net), Jul 06 2003
%I A085980
%S A085980 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A085980 0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A085980 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0
%N A085980 Number of 6's in decimal expansion of prime(n).
%Y A085980 Cf. 0's A085974, 1's A085975, 2's A085976, 3's A085977, 4's A085978, 5's A085979, 7's A085981, 8's A085982, 9's A085983.
%Y A085980 Adjacent sequences: A085977 A085978 A085979 this_sequence A085981 A085982 A085983
%Y A085980 Sequence in context: A011733 A011732 A011731 this_sequence A023974 A011730 A011729
%K A085980 base,nonn
%O A085980 1,1
%A A085980 Jason Earls (jcearls(AT)kskc.net), Jul 06 2003
%I A071031
%S A071031 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A071031 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
%U A071031 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0
%N A071031 Triangle giving successive states of cellular automaton generated by "rule 62".
%C A071031 Row n has length 2n+1.
%D A071031 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071031 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%Y A071031 Adjacent sequences: A071028 A071029 A071030 this_sequence A071032 A071033 A071034
%Y A071031 Sequence in context: A011730 A011729 A011728 this_sequence A011727 A088918 A011726
%K A071031 nonn,tabf
%O A071031 0,1
%A A071031 Hans Havermann (hahaj(AT)rogers.com), May 26 2002
%I A023973
%S A023973 0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A023973 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,
%U A023973 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A023973 First bit in fractional part of binary expansion of 6-th root of n.
%t A023973 Array[ Function[ n, RealDigits[ N[ Power[ n,1/6 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023973 Adjacent sequences: A023970 A023971 A023972 this_sequence A023974 A023975 A023976
%Y A023973 Sequence in context: A045701 A011725 A037808 this_sequence A044941 A011724 A037807
%K A023973 nonn,base
%O A023973 1,1
%A A023973 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A025465
%S A025465 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,
%T A025465 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%U A025465 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0
%N A025465 Partitions of n into 3 distinct nonnegative cubes.
%Y A025465 Adjacent sequences: A025462 A025463 A025464 this_sequence A025466 A025467 A025468
%Y A025465 Sequence in context: A037807 A037817 A025468 this_sequence A044940 A037825 A073490
%K A025465 nonn
%O A025465 0,1
%A A025465 David W. Wilson (davidwwilson(AT)comcast.net)
%I A023972
%S A023972 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,
%T A023972 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A023972 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1
%N A023972 First bit in fractional part of binary expansion of 5-th root of n.
%t A023972 Array[ Function[ n, RealDigits[ N[ Power[ n,1/5 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023972 Adjacent sequences: A023969 A023970 A023971 this_sequence A023973 A023974 A023975
%Y A023972 Sequence in context: A011713 A011699 A011710 this_sequence A044937 A025459 A079365
%K A023972 nonn,base
%O A023972 1,1
%A A023972 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A023971
%S A023971 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A023971 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A023971 1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A023971 First bit in fractional part of binary expansion of 4-th root of n.
%t A023971 Array[ Function[ n, RealDigits[ N[ Power[ n,1/4 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023971 Adjacent sequences: A023968 A023969 A023970 this_sequence A023972 A023973 A023974
%Y A023971 Sequence in context: A011670 A011666 A011669 this_sequence A079260 A025457 A080343
%K A023971 nonn,base
%O A023971 1,1
%A A023971 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A080584
%S A080584 0,0,0,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
%T A080584 1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,
%U A080584 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A080584 A run of 3*2^n 0's followed by a run of 3*2^n 1's, for n=0, 1, 2, ...
%F A080584 a(n) = (1 - (-1)^A079944(A002264(n)) )/2, A079944(A002264(n))=floor(log[2](4*(floor((n+6)/3))/3)) - floor(log[2](floor((n+6)/3))) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003
%F A080584 Also a(n) = A079944(A002264(n)) = floor(log[2](4*(floor((n+6)/3))/3)) - floor(log[2](floor((n+6)/3))) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003
%p A080584 f:=(c,n)->seq(c,i = 1..3*2^n); [f(0,0),f(1,0),f(0,1),f(1,1),f(0,2),f(1,2),f(0,3),f(1,3)]; f;
%Y A080584 Equals A080586 - 1.
%Y A080584 Adjacent sequences: A080581 A080582 A080583 this_sequence A080585 A080586 A080587
%Y A080584 Sequence in context: A064911 A066247 A011658 this_sequence A023970 A072626 A011659
%K A080584 nonn
%O A080584 0,1
%A A080584 njas, Feb 23 2003
%I A023970
%S A023970 0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,
%T A023970 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,
%U A023970 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1
%N A023970 First bit in fractional part of binary expansion of cube root of n.
%t A023970 Array[ Function[ n, RealDigits[ N[ Power[ n,1/3 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023970 Adjacent sequences: A023967 A023968 A023969 this_sequence A023971 A023972 A023973
%Y A023970 Sequence in context: A066247 A011658 A080584 this_sequence A072626 A011659 A056029
%K A023970 nonn,base
%O A023970 1,1
%A A023970 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A072609
%S A072609 0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,
%T A072609 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,
%U A072609 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A072609 Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.
%F A072609 a(n)=Mod[A004648(n),2]*Mod[A004648(n+1),2]= A072608(n)*A072608(n+1)
%e A072609 Take n = 11,12,13,14: A004648[n]=9,1,2,1. Parity A072608(n) = 1,1,0,1. So ..11.. transforms into 01 between n = 11 and n = 12: a(11) = 1, a(12)=0. With increasing n, A072609(n) changes from ..0000.. into ...1111. reflected by this sequence. by a range consisting only of 1-s. This secondary alternation also goes on.
%t A072609 mm[x_] :=Mod[Mod[Prime[x],x],2] Table[mm[w]*mm[w+1],{w,1,256}]
%Y A072609 Cf. A004648, A072608.
%Y A072609 Adjacent sequences: A072606 A072607 A072608 this_sequence A072610 A072611 A072612
%Y A072609 Sequence in context: A072626 A011659 A056029 this_sequence A025455 A025125 A059648
%K A072609 nice,nonn
%O A072609 1,1
%A A072609 Labos E. (labos(AT)ana1.sote.hu), Jun 24 2002
=============================================================================================
For lookup 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368
I got:
Transformation T109 gave a match with:
%I A011742
%S A011742 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A011742 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,
%U A011742 1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1
%N A011742 A binary m-sequence: expansion of reciprocal of x^29+x^2+1.
%D A011742 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
%D A011742 H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
%D A011742 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
%Y A011742 Adjacent sequences: A011739 A011740 A011741 this_sequence A011743 A011744 A011745
%Y A011742 Sequence in context: A011745 A011744 A011743 this_sequence A011741 A011740 A085974
%K A011742 nonn
%O A011742 0,1
%A A011742 njas
%I A060510
%S A060510 0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,
%T A060510 1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,
%U A060510 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1
%N A060510 Alternating with hexagonal stutters: if n is hexagonal (2k^2-k i.e. A000384) then a(n)=a(n-1), otherwise a(n)=1-a(n-1).
%F A060510 a(n) =A002262(n) mod 2 =A060511(n) mod 2.
%e A060510 Hexagonal numbers start 1,6,15, ... so this sequence goes 0 0 (stutter at 1) 1 0 1 0 0 (stutter at 6) 1 0 1 0 1 0 1 0 0 (stutter at 15) 1 0 etc.
%Y A060510 As a simple triangular or square array virtually the only sequences which appear are A000004, A000012 and A000035.
%Y A060510 Adjacent sequences: A060507 A060508 A060509 this_sequence A060511 A060512 A060513
%Y A060510 Sequence in context: A028999 A014707 A059125 this_sequence A072629 A022925 A051840
%K A060510 easy,nonn,tabl
%O A060510 0,1
%A A060510 Henry Bottomley (se16(AT)btinternet.com), Mar 22 2001
%I A072629
%S A072629 0,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
%T A072629 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A072629 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A072629 Parity of n*Foor[Log[n]].
%F A072629 a(n)= Mod[n*Floor[Log[n]],2]
%e A072629 Parity either alternates or it is steadily 0. Intervals of such kind also change and return: 01010...0000....0101..etc.
%Y A072629 See also A004648, A072608, A072609, A072610, A072630.
%Y A072629 Adjacent sequences: A072626 A072627 A072628 this_sequence A072630 A072631 A072632
%Y A072629 Sequence in context: A014707 A059125 A060510 this_sequence A022925 A051840 A056051
%K A072629 nonn
%O A072629 1,1
%A A072629 Labos E. (labos(AT)ana1.sote.hu), Jun 28 2002
%I A000035 M0001
%S A000035 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,
%T A000035 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
%U A000035 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A000035 A simple periodic sequence.
%C A000035 Least significant bit of n, lsb(n).
%C A000035 Also decimal expansion of 1/99.
%D A000035 A. K. Whitford, Binet's Formula Generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
%H A000035 Y. Puri and T. Ward, <a href="http://www.math.uwaterloo.ca/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000035 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000035 a(n)={1 - (-1)^n}/2. a(n) = n mod 2.
%F A000035 Multiplicative with a(p^e) = p%2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000035 G.f.: x/(1-x^2). E.g.f.: sinh(x). a(n)=n mod 2. a(n)=1/2 - (-1)^n/2. - Paul Barry (pbarry(AT)wit.ie), Mar 11 2003
%F A000035 a(n)=(A000051(n)-A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
%p A000035 A000035:=n->n mod 2;
%p A000035 [ seq(i mod 2, i=0..100) ];
%o A000035 (PARI) a(n)=n%2
%Y A000035 Ones complement of A059841. Cf. A053644 for most significant bit.
%Y A000035 Adjacent sequences: A000032 A000033 A000034 this_sequence A000036 A000037 A000038
%Y A000035 Sequence in context: A073424 A082848 A061265 this_sequence A071029 A071030 A073445
%K A000035 core,easy,nonn,nice,mult
%O A000035 0,1
%A A000035 njas
%I A071029
%S A071029 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,
%T A071029 1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,
%U A071029 1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A071029 Triangle giving successive states of cellular automaton generated by "rule 22".
%C A071029 Row n has length 2n+1.
%D A071029 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071029 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%Y A071029 Adjacent sequences: A071026 A071027 A071028 this_sequence A071030 A071031 A071032
%Y A071029 Sequence in context: A082848 A061265 A000035 this_sequence A071030 A073445 A082446
%K A071029 nonn,tabf
%O A071029 0,1
%A A071029 Hans Havermann (hahaj(AT)rogers.com), May 26 2002
%I A071030
%S A071030 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,
%T A071030 0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,
%U A071030 1,0,0,0,1,0,0,0,1,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0
%N A071030 Triangle giving successive states of cellular automaton generated by "rule 54".
%C A071030 Row n has length 2n+1.
%D A071030 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071030 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%Y A071030 Adjacent sequences: A071027 A071028 A071029 this_sequence A071031 A071032 A071033
%Y A071030 Sequence in context: A061265 A000035 A071029 this_sequence A073445 A082446 A071024
%K A071030 nonn,tabf
%O A071030 0,1
%A A071030 Hans Havermann (hahaj(AT)rogers.com), May 26 2002
%I A072608
%S A072608 0,1,0,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,
%T A072608 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,
%U A072608 1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A072608 Parity of remainder Mod[p(n),n]=A004648(n).
%F A072608 a(n)=Mod[Mod[p(n),n],2]=Mod[A004648(n),2]
%e A072608 n=25:p(25)=97,Mod[97,25]=22, a(25)=Mod[22,2]=0. With increasing n, a(n) alternates:...010101..,followed after by a range consisting only of 1-s. This secondary alternation also goes on.
%t A072608 mm[x_] :=Mod[Mod[Prime[x],x],2] Table[mm[w],{w,1,256}]
%Y A072608 Cf. A004648.
%Y A072608 Adjacent sequences: A072605 A072606 A072607 this_sequence A072609 A072610 A072611
%Y A072608 Sequence in context: A022924 A023532 A072165 this_sequence A030301 A071981 A057212
%K A072608 nice,nonn
%O A072608 1,1
%A A072608 Labos E. (labos(AT)ana1.sote.hu), Jun 24 2002
%I A016141
%S A016141 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,
%T A016141 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A016141 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%V A016141 1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,
%W A016141 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%X A016141 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A016141 Inverse of 2132th cyclotomic polynomial.
%p A016141 with(numtheory,cyclotomic); c:=n->series(1/cyclotomic(n,x),x,80);
%Y A016141 Adjacent sequences: A016138 A016139 A016140 this_sequence A016142 A016143 A016144
%Y A016141 Sequence in context: A014845 A014757 A014581 this_sequence A016245 A015933 A015621
%K A016141 sign
%O A016141 0,1
%A A016141 Simon Plouffe (plouffe(AT)math.uqam.ca)
%I A016245
%S A016245 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,
%T A016245 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A016245 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%V A016245 1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,
%W A016245 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%X A016245 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A016245 Inverse of 2236th cyclotomic polynomial.
%p A016245 with(numtheory,cyclotomic); c:=n->series(1/cyclotomic(n,x),x,80);
%Y A016245 Adjacent sequences: A016242 A016243 A016244 this_sequence A016246 A016247 A016248
%Y A016245 Sequence in context: A014757 A014581 A016141 this_sequence A015933 A015621 A015517
%K A016245 sign
%O A016245 0,1
%A A016245 Simon Plouffe (plouffe(AT)math.uqam.ca)
%I A015933
%S A015933 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,
%T A015933 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A015933 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1
%V A015933 1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,
%W A015933 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%X A015933 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,1,0,-1
%N A015933 Inverse of 1924th cyclotomic polynomial.
%p A015933 with(numtheory,cyclotomic); c:=n->series(1/cyclotomic(n,x),x,80);
%Y A015933 Adjacent sequences: A015930 A015931 A015932 this_sequence A015934 A015935 A015936
%Y A015933 Sequence in context: A014581 A016141 A016245 this_sequence A015621 A015517 A015205
%K A015933 sign
%O A015933 0,1
%A A015933 Simon Plouffe (plouffe(AT)math.uqam.ca)
and T108 transform didn't match against anything with either of the two lookups above.
==================================================================================
For lookup 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584
4181 6765 10946
Transformation T108 gave a match with:
%I A001045 M2482 N0983
%S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,
%T A001045 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,
%U A001045 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531
%N A001045 Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
%C A001045 Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square tiles.
%C A001045 Also the number of ways to tie a necktie using n+2 turns. So three turns make an "oriental", four make a "four in hand", and for 5 turns there are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises from a special random walk on a triangular grid with side conditions (see Fink and Mao, 1999). - arne.ring(AT)epost.de, Mar 18 2001
%C A001045 Also the number of compositions of n+1 ending with an odd part (a(2)=3 because 3, 21, 111 are the only compositions of 3 ending with an odd part). Also the number of compositions of n+2 ending with an even part (a(2)=3 because 4, 22, 112 are the only compositions of 4 ending with an even part). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2001
%C A001045 Arises in study of sorting by merge insertions and in analysis of a method for computing GCDs - see Knuth reference.
%C A001045 Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002: Number of perfect matchings of a 2 X n grid upon replacing unit squares with tetrahedra (C_4 to K_4):
%C A001045 o----o----o----o...
%C A001045 | \/ | \/ | \/ |
%C A001045 | /\ | /\ | /\ |
%C A001045 o----o----o----o...
%C A001045 Also the numerators of the reduced fractions in the alternating sum 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + ... - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Feb 07 2002
%C A001045 Also, if A(n),B(n),C(n) are the angles of the n-orthic triangle of ABC then A(1) = Pi - 2A, A(n) = s(n)Pi + (-2)^nA where s(n) = (-1)^(n-1) * a(n) [1-orthic triangle = the orthic triangle of ABC, n-orthic triangle = the orthic triangle of the (n-1)-orthic triangle] - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 05 2002
%C A001045 Also the number of words of length n+1 in the two letters s and t that reduce to the identity 1 by using the relations sss=1, tt=1, and stst=1. The generators s and t and the three stated relations generate the group S3. - John W. Layman (layman(AT)math.vt.edu), Jun 14 2002
%C A001045 Sums of pair of consecutive terms give all powers of 2 in increasing order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 15 2002
%C A001045 Excess clockwise moves (over anti-clockwise) needed to move a tower of size n to the clockwise peg is -(-1)^n(2^n - (-1)^n)/3; a(n)=its unsigned version. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 01 2002
%C A001045 Also the absolute value of the number represented in base -2 by the string of n 1's, the negabinary repunit. The Mersenne numbers (A000225 and its subsequences) are the binary repunits. - Rick L. Shepherd(AT)prodigy.net (rshepherd2(AT)hotmail.com), Sep 16 2002
%C A001045 Note that 3a(n)+(-1)^n=2^n is significant for Pascal' triangle A007318. It arises from a Jacobsthal decomposition of Pascal's triangle illustrated by 1+7+21+35+35+21+7+1 = (7+35+1)+(1+35+7)+(21+21) = 43 + 43 + 42 = 3a(7)-1; 1+8+28+56+70+56+29+8+1 = (1+56+28)+(28+56+1)+(8+70+8) = 85 + 85 + 86 = 3a(8)+1. - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%C A001045 If (m + n) is odd, then (3*(a(m) + a(n)) is always of the form a^2 + 2*b^2, where a and b both equal powers of 2; consequently every factor of (a(m) + a(n)) is always of the form a^2 + 2*b^2. - Matthew Vandermast (ghodges14(AT)msn.com), Jul 12 2003
%C A001045 Number of positive integers requiring exactly n signed bits in the non-adjacent form representation.
%C A001045 Counts walks between adjacent vertices of a triangle - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
%F A001045 a(n) = 2^(n-1) - a(n-1). a(n) = 2*a(n-1) - (-1)^n = {2^n - (-1)^n}/3.
%F A001045 G.f.: x/(1-x-2*x^2). E.g.f.: (exp(2*x)-exp(-x))/3.
%F A001045 a(2n)=2*a(2n-1)-1 for n>=1, a(2n+1)=2*a(2n)+1 for n>=0. - Lee Hae-hwang (mathmaniac(AT)empal.com), Oct 11 2002; corrected by Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 Also a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y)=xF(n-1)(x,y)+yF(n-2)(x,y), with y=2x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 a(n)=Sum{k=0..floor(n,3), binomial(n,f(n-1)+3k)} a(n)=Sum(k=0..floor(n,3), binomial(n,f(n-2)+3k}, where f(n)=(0,2,1,0,2,1,...)=A080424(n). - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%F A001045 a(n)=sum{k=1..n, binomial(n,k)(-1)^(n+k)*3^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), Apr 02 2003
%F A001045 The ratios a(n)/2^(n-1) converge to 2/3, and every fraction after 1/2 is the arithmetic mean of the two preceding fractions. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 05 2003
%F A001045 a(n)=U(n-1,i/(2sqrt(2)))(-isqrt(2))^(n-1) with i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
%e A001045 a(2) = 3 because the tiling of the 3x2 rectangle has either only 1 X 1 tiles, or one 2 X 2 tile in one of two positions (together with 2 1 X 1 tiles)
%o A001045 (PARI) a(n)=if(n<0,0,(2^n-(-1)^n)/3)
%Y A001045 Partial sums of this sequence give A000975, where there are additional comments from B. E. Williams and Bill Blewett on the tie problem. Cf. A049883, A026644.
%Y A001045 A002487(A001045(n))=A000045(n).
%Y A001045 Row sums of A059260. Equals A026644(n) + 1 for n > 1.
%Y A001045 a(n)= A073370(n-1,0), n>=1 (first column of triangle).
%Y A001045 Apart from initial term, equals A026644(n+1) + 1.
%Y A001045 See also A081857.
%Y A001045 Adjacent sequences: A001042 A001043 A001044 this_sequence A001046 A001047 A001048
%Y A001045 Sequence in context: A074710 A057460 A045515 this_sequence A077925 A084230 A077465
%K A001045 nonn,nice,easy
%O A001045 0,4
%A A001045 njas
%E A001045 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 23 1999. Zerr reference from Len Smiley (smiley(AT)math.uaa.alaska.edu), May 21 2001.
and
Transformation T109 gave a match with:
%I A023976
%S A023976 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A023976 0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A023976 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A023976 First bit in fractional part of binary expansion of 9-th root of n.
%t A023976 Array[Function[n, RealDigits[N[Power[n,1/9 ],10 ],2 ]// (#[[1,#[[2 ] ]+1 ] ])& ],110 ]
%Y A023976 Adjacent sequences: A023973 A023974 A023975 this_sequence A023977 A023978 A023979
%Y A023976 Sequence in context: A000004 this_sequence A025469 A025466 A072769
%K A023976 nonn,base
%O A023976 1,1
%A A023976 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A011745
%S A011745 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A011745 0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A011745 1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A011745 A binary m-sequence: expansion of reciprocal of x^32+x^28+x^27+x+1.
%D A011745 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
%D A011745 H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
%D A011745 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
%Y A011745 Adjacent sequences: A011742 A011743 A011744 this_sequence A011746 A011747 A011748
%Y A011745 Sequence in context: A025466 A072769 A070204 this_sequence A011744 A011743 A011742
%K A011745 nonn
%O A011745 0,1
%A A011745 njas
%I A023975
%S A023975 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,
%T A023975 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A023975 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A023975 First bit in fractional part of binary expansion of 8-th root of n.
%t A023975 Array[ Function[ n, RealDigits[ N[ Power[ n,1/8 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023975 Adjacent sequences: A023972 A023973 A023974 this_sequence A023976 A023977 A023978
%Y A023975 Sequence in context: A011740 A085974 A011739 this_sequence A011738 A011737 A076142
%K A023975 nonn,base
%O A023975 1,1
%A A023975 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A011735
%S A011735 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,
%T A011735 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,
%U A011735 1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1
%N A011735 A binary m-sequence: expansion of reciprocal of x^22+x+1.
%D A011735 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
%D A011735 H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
%D A011735 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
%Y A011735 Adjacent sequences: A011732 A011733 A011734 this_sequence A011736 A011737 A011738
%Y A011735 Sequence in context: A076142 A011736 A085982 this_sequence A011734 A011733 A011732
%K A011735 nonn
%O A011735 0,1
%A A011735 njas
%I A023974
%S A023974 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A023974 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A023974 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A023974 First bit in fractional part of binary expansion of 7-th root of n.
%t A023974 Array[ Function[ n, RealDigits[ N[ Power[ n,1/7 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023974 Adjacent sequences: A023971 A023972 A023973 this_sequence A023975 A023976 A023977
%Y A023974 Sequence in context: A011732 A011731 A085980 this_sequence A011730 A011729 A011728
%K A023974 nonn,base
%O A023974 1,1
%A A023974 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A023973
%S A023973 0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A023973 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,
%U A023973 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A023973 First bit in fractional part of binary expansion of 6-th root of n.
%t A023973 Array[ Function[ n, RealDigits[ N[ Power[ n,1/6 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023973 Adjacent sequences: A023970 A023971 A023972 this_sequence A023974 A023975 A023976
%Y A023973 Sequence in context: A045701 A011725 A037808 this_sequence A044941 A011724 A037807
%K A023973 nonn,base
%O A023973 1,1
%A A023973 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A023972
%S A023972 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,
%T A023972 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A023972 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1
%N A023972 First bit in fractional part of binary expansion of 5-th root of n.
%t A023972 Array[ Function[ n, RealDigits[ N[ Power[ n,1/5 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023972 Adjacent sequences: A023969 A023970 A023971 this_sequence A023973 A023974 A023975
%Y A023972 Sequence in context: A011713 A011699 A011710 this_sequence A044937 A025459 A079365
%K A023972 nonn,base
%O A023972 1,1
%A A023972 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A023971
%S A023971 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A023971 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A023971 1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A023971 First bit in fractional part of binary expansion of 4-th root of n.
%t A023971 Array[ Function[ n, RealDigits[ N[ Power[ n,1/4 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023971 Adjacent sequences: A023968 A023969 A023970 this_sequence A023972 A023973 A023974
%Y A023971 Sequence in context: A011670 A011666 A011669 this_sequence A079260 A025457 A080343
%K A023971 nonn,base
%O A023971 1,1
%A A023971 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A080584
%S A080584 0,0,0,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
%T A080584 1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,
%U A080584 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A080584 A run of 3*2^n 0's followed by a run of 3*2^n 1's, for n=0, 1, 2, ...
%F A080584 a(n) = (1 - (-1)^A079944(A002264(n)) )/2, A079944(A002264(n))=floor(log[2](4*(floor((n+6)/3))/3)) - floor(log[2](floor((n+6)/3))) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003
%F A080584 Also a(n) = A079944(A002264(n)) = floor(log[2](4*(floor((n+6)/3))/3)) - floor(log[2](floor((n+6)/3))) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003
%p A080584 f:=(c,n)->seq(c,i = 1..3*2^n); [f(0,0),f(1,0),f(0,1),f(1,1),f(0,2),f(1,2),f(0,3),f(1,3)]; f;
%Y A080584 Equals A080586 - 1.
%Y A080584 Adjacent sequences: A080581 A080582 A080583 this_sequence A080585 A080586 A080587
%Y A080584 Sequence in context: A064911 A066247 A011658 this_sequence A023970 A072626 A011659
%K A080584 nonn
%O A080584 0,1
%A A080584 njas, Feb 23 2003
%I A023970
%S A023970 0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,
%T A023970 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,
%U A023970 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1
%N A023970 First bit in fractional part of binary expansion of cube root of n.
%t A023970 Array[ Function[ n, RealDigits[ N[ Power[ n,1/3 ],10 ],2 ]// (#[ [ 1,#[ [ 2 ] ]+1 ] ])& ],110 ]
%Y A023970 Adjacent sequences: A023967 A023968 A023969 this_sequence A023971 A023972 A023973
%Y A023970 Sequence in context: A066247 A011658 A080584 this_sequence A072626 A011659 A056029
%K A023970 nonn,base
%O A023970 1,1
%A A023970 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
But neither
%S A019590 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A019590 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A019590 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A019590 Fermat's Last Theorem: a(n) = 1 if x^n+y^n=z^n has a nontrivial solution in integers, otherwise a(n) = 0.
%K A019590 nonn,nice,easy
%O A019590 1,1
or A000129 occur anywhere!
List of transformations used:
T108 invert: define b by 1+SUM b(n)x^n = 1/(1 - SUM a(n)x^n)
T109 invert: define b by 1+SUM a(n)x^n = 1/(1 - SUM b(n)x^n)
(and I think we should have T109 matching to A019590 and T108
matching to A000129, if the offsets were correctly chosen. But no!)
Not any better if I'll try
lookup 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
or:
lookup 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210
or:
lookup 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210
none of them matches to A000045.
So, what it is going on here!?
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