# Duplicate hunting, cont.

Andrew Plewe aplewe at sbcglobal.net
Wed Apr 18 02:32:31 CEST 2007

```Dear Andrew,
You was find quite a few from thousends. Congratulations!
A029549 </%7Enjas/sequences/A029549> and  A076715 (not  A123481 which
doesn't exists)

BEST WISHES
ARTUR

Andrew Plewe napisał(a):
> Straightforward duplicates:
>
> A023727 and A043354
> A087921 and A087922
> A101294 and A101935
> A056209 and A056157 and (maybe?) A098671
> A056266 and A101019
> A056252 and A101829
> A057130 and A121404
> A090484 and A090489
> A086749 and A086750
> A076671 and A076718
> A077491 and A088401
>
> Possibly the same sequence:
>
> A123481 and A076715 (can this really just be coincidence?)
> A107317 and A112667
>
> 	-Andrew Plewe-
>
>
>
> __________ NOD32 Informacje 2199 (20070417) __________
>
> Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
> http://www.nod32.com lub http://www.nod32.pl
>
>
>
>

Define a sequence as follows:

a(0) and a(1) = nonnegative integers.

For n >= 2,
a(n) = the number of terms among {a(0),a(1),...,a(n-1)} which are equal
to (a(n-1)+a(n-2)).

So, for example:

(I may have made a mistake or two.)

a(0)=0, a(1)=1:

0,1,1,0,2,1,0,3,1,0,4,1,0,5,...

As you see, this sequence extends upwards forever with the pattern
{..1,0,m,1,0,m+1,1,0,m+2,...}.

On the other hand, we can have the sequence starting with a(0) =1, a(1)=2:

1,2,0,1,2,0,2,3,0,1,3,0,2,4,0,1,4,0,2,...

This sequence has the pattern
{..m,0,1,m,0,2,m+1,0,1,m+1,0,2,m+2,0,1,m+2,0,2,...}.

But the majority of sequences seems to be of this type:
(Although, many go on for a little while before hitting the string of
1's.)

a(0)=2, a(1)=0.

2,0,1,1,1,1,1,1,1,....

Of course, the rest of the sequence is made up of only 1's.

An interesting sequence which repeats after some point, and so is
bounded, is:

a(0)= a(1) = 4.

4,4,0,2,1,0,1,2,0,2,3,0,1,3,2,0,4,3,0,3,4,0,4,5,0,1,4,1,1,4,1,1,4,1,1,4,1,.
..

After some point, the sequence is just the repetition of {1,4,1}.

I wonder what the plot on a grid would look like where a grid-square is
filled in with, say, black (inspired by the Mandelbrot set) if the
another color otherwise.

I get the start of such a plot looking like this:
(I may have very well made an error.)

a(1)
|
V
0 . . * * * *
1 . . . * * *
2 . . . * * *
3 * * * ? * *
4 * * * * * *
5 * * * * * ?

. = sequence increases without bound.
* = sequence repeats itself after some point.
? = I don't know.

Can anybody add anything to this discussion?

I am choosing not to personally submit any of these sequences to the OEIS
because they are too predictable after some point, at least with most

Thanks,
Leroy Quet

I think a "trickle" approach is useful; overwhelming people with data
generally results in no one looking at it. I used a partly automated process
to compile the list (i.e., import sequences & A numbers into sql database,
treat sequences as lexical strings, make table of duplicate strings grouped
by A number, export table as flat file) which took about 20 minutes.
Reviewing the list, however, and getting something useful out of it is a
different animal that requires time and human attention. As I said in a
previous email, I'm filtering out a large number of sequences by the same
author because they'd otherwise drown out the more useful (for now)
duplicates. There are also many that have the same terms but will differ at

On a related note, perhaps there could be a keyword for "possible"
duplicates, i.e. the finder or editor doesn't know for sure, perhaps someone
could provide a proof. A connotation of it's use is the duplication is
"interesting". With that, here are a few for today:

A055490 and A126783 ([A126783] (n!-1) mod (n-1)! only produces numbers of
the form (n-1)! - 1 [A055490] for integer values of n)
A063448 and A105724 ([A105724] sqrt(pi^2 + pi^2) = sqrt(2pi^2) = pi *
A072215 and A106170
A108163 and A108171

Possible duplicates:

A073870 and A085088 (A085088 isn't strictly defined, could be multiple
interepretations)
A025017 and A102043
A093581 and A102557 (this is what I'd consider a candidate for a "possible"
duplicate keyword)

-----Original Message-----

Maximilian said:

> [the] truth of the
> correspondence can easily be checked by a script (at least for
> "standard cases" where each sequence has at least say 10 terms, and
> all terms match)

this proves nothing.  Remember that the OEIS is a scientific
reference work.  You should have a proof before you claim
that two sequences are identical.

Of course in the majority of cases the proof is obvious,
as in this pair found by Andrew:

> A123481 and A076715

(I am merging them)

On the other hand, for this pair that he found:

> A107317 and A112667

it is less obvious that they are the same (although they probably
are), and I am leaving them as distinct entries for the moment,

All the other pairs that Andrew found will be merged at the next update -
thanks, Andrew!

Best regards

Neil Sloane

A076715  a(n+3) = 35*a(n+2) - 35*a(n+1) + a(n).

is the same as

A029549  Triangular numbers that are twice other triangular numbers.

except the initial 0 term.

---------- Original Message ----------------------------------

>
>Possibly the same sequence:
>
>A123481 and A076715 (can this really just be coincidence?)
>
>	-Andrew Plewe-
>
>
>

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Leroy Quet <qq-quet at mindspring.com> wrote:
:Define a sequence as follows:
:
:a(0) and a(1) = nonnegative integers.
:
:For n >= 2,
:a(n) = the number of terms among {a(0),a(1),...,a(n-1)} which are equal
:to (a(n-1)+a(n-2)).
[...]
:I wonder what the plot on a grid would look like where a grid-square is
:filled in with, say, black (inspired by the Mandelbrot set) if the
:sequence relating to its coordinates is bounded, and is filled in with
:another color otherwise.
:
:I get the start of such a plot looking like this:
:(I may have very well made an error.)
:
:    a(0) ->
:a(1)
:|
:V
:  0 1 2 3 4 5
:0 . . * * * *
:1 . . . * * *
:2 . . . * * *
:3 * * * ? * *
:4 * * * * * *
:5 * * * * * ?
:
:. = sequence increases without bound.
:* = sequence repeats itself after some point.
:? = I don't know.

Using this to get a quick list of the first 1000 terms of any such sequence:

I get (3, 3): 3 3 0 2 1 2 2 0 3 3 0 4 1 0 2 4 0 2 5 0 1 3 2 1 5 0 2 7 0 1 5 0
3 6 0 1 6 1 1 7 0 2 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]

(5, 5): 5 5 0 2 1 0 1 2 0 2 3 2 2 0 5 3 0 2 6 0 1 3 0 3 4 0 1 4 3 0 5 4 0 3 6
0 2 7 0 1 5 2 1 6 1 1 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]

Giving:
0 . . * * * *
1 . . . * * *
2 . . . * * *
3 * * * . * *
4 * * * * * *
5 * * * * * .

I think there may be some interesting maths involved in characterising
the behaviour of the 3-valued function f(a(0), a(1), n), I agree that
there doesn't currently seem much of an interesting sequence here.

Hugo

hv at crypt.org wrote:
>...
>I get (3, 3): 3 3 0 2 1 2 2 0 3 3 0 4 1 0 2 4 0 2 5 0 1 3 2 1 5 0 2 7 0 1 5 0
>3 6 0 1 6 1 1 7 0 2 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]
>
>(5, 5): 5 5 0 2 1 0 1 2 0 2 3 2 2 0 5 3 0 2 6 0 1 3 0 3 4 0 1 4 3 0 5 4 0 3 6
>0 2 7 0 1 5 2 1 6 1 1 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]
>
>Giving:
>  0 1 2 3 4 5
>0 . . * * * *
>1 . . . * * *
>2 . . . * * *
>3 * * * . * *
>4 * * * * * *
>5 * * * * * .
>
>I think there may be some interesting maths involved in characterising
>the behaviour of the 3-valued function f(a(0), a(1), n), I agree that
>there doesn't currently seem much of an interesting sequence here.
>
>Hugo

I was hoping for some interesting pattern of .'s and *'s when the above
plot is taken to many more terms. But I conjecture, sadly, that the only
.'s in the infinite grid are those represented above, the rest of the
grid-squares being of only *'s.

(Again: "*" represents a (a(0),a(1)) which leads to a sequence which
repeats after some point. "." represents a (a(0),a(1)) which leads to a

Thanks,
Leroy Quet

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