Joshua Zucker joshua.zucker at gmail.com
Fri Feb 20 17:56:41 CET 2009

```On Thu, Feb 19, 2009 at 12:51 PM, vincenzo.librandi at tin.it
<vincenzo.librandi at tin.it> wrote:
> %C A001105 Except the first term, numbers n
> such that (9*n^3)/8 is a
> square. Example: (9*2^3)/8=9=3^2; (9*8^3)
> /8=576=24^2;
> (9*18^3)/8=6561=81^2; (9*32^3)/8=36864=192^2; [From
> Vincenzo Librandi
> (vincenzo.librandi(AT)tin.it), Feb 16 2009]

Your comment says that 9*n^3 / 8 is a square.

But 9*n^3 / 8 is a square iff n^3/8 is a square iff n^3/2 is a square
and n is even iff n is of the form 2*k^2 for some k, and the sequence
is DEFINED as being 2*k^2, so your observation is trivial.

Also the "except for the first term" is unnecessary, since 0 is a square also.

Is there some significance to 9*n^3/8 that I'm missing?  I mean, we
could just as well say "numbers n such that 729 * n^17 / 262144 is a
square" or something like that?

> %C
> A001081 Except the first term of [A001080] and of [A001081], If
> X=
> [A001081] (1,8,127,2024,32257,..,); Y=[A001080]
> (0,3,48,765,1192,..,)
> and A=[A010727] (7,7,7,..,) we have, for all
> terms, Pell's equation X^2-
> A*Y^2=1. Example: 8^2-7*3^2=1;
> 127^2-7*48^2=1; 2024^2-7*765^2=1;
> 32257^2-7*12192^2=1; [From Vincenzo
> Librandi (vincenzo.librandi(AT)tin.
> it), Feb 16 2009]
> %Y A001081 Cf. A001080, A010727 [From Vincenzo
> Librandi
> (vincenzo.librandi(AT)tin.it), Feb 16 2009]

The "except the first term" is unnecessary again, and the constant
sequence A010727 could simply be replaced by the number 7 for more
clarity.

That is, I would just say

%C A001081 (A001081)^2 - 7*(A001080)^2 = 1.

I do think this is a useful comment, because the connection between
the recursive formula here and the Pell equation is well-known but
should be noted in these sequences!  (My comments here also apply to
A001080 of course).

Also, I note that the definition of A001081 is not sufficient.  It says
%N A001081 a(n) = 16a(n-1) - a(n-2).
but it should say
%N A001081 a(n) = 16a(n-1) - a(n-2), with a(0) = 1 and a(1) = 8.

> %F A011944 a(n)
> =14*a(n-1)-a(n-2), with a(1)=0, a(2)=2: Example:
> a(3)=14*a(2)-a(1)=14*2-
> 0=28; a(4)=14*28-2=390; a(5)=14*390-28=5432.
> [From Vincenzo Librandi
> (vincenzo.librandi(AT)tin.it), Feb 16 2009]

Isn't this already the definition of A011944?  This comment seems redundant.

> %C A010727 Except the
> first term of [A001080] and of [A001081], If
> X=[A001081]
> (1,8,127,2024,32257,..,); Y=[A001080]
> (0,3,48,765,1192,..,) and A=
> [A010727] (7,7,7,..,) we have, for all
> terms, Pell's equation X^2-
> A*Y^2=1. Example: 8^2-7*3^2=1;
> 127^2-7*48^2=1; 2024^2-7*765^2=1;
> 32257^2-7*12192^2=1; [From Vincenzo
> Librandi (vincenzo.librandi(AT)tin.
> it), Feb 16 2009]

If every reference to the number 7 in the OEIS merited a reference to
the constant sequence of 7s, this entry would quickly become unusable
I should think!

>
> %C A047522 Except
> for the first term of [A047522] ad the first term of
> [A074378], if X=
> [A047522], Y=[A010709], A=[A074378], we have, for all
> other terms,
> Pell's equation X^2-A*Y^2=1. Example 9^2-5*4^2=1;
> 15^2-14*4^2=1; 17^2-
> 18*4^2=1 [From Vincenzo Librandi
> (vincenzo.librandi(AT)tin.it), Feb 14
> 2009]
> %Y A047522 Cf. A010709, A074378 [From Vincenzo Librandi
> (vincenzo.
> librandi(AT)tin.it), Feb 14 2009]

The reference to the constant sequence A010709 is unnecessary and
should be replaced by the number 4.  Also it is true for the first
term, so the "except for the first term" part should be deleted as
well.  That is, all you need to write for this comment is

%C A047522 (A047552)^2 - 16*A074378 = 1.

>  %C A078371 Except the two terms of
> [A141530] ad the first term of
> [A046092[, if X=[A141530], A=[A078371],
> Y=[A046092], we have, for all
> others terms, Pell's equation: X^2-
> A*Y^2=1. Example: 9^2-5*4^2=1;
> 55^2-21*12^2=1; 161^2-45*24^2=1. [From
> Vincenzo Librandi
> (vincenzo.librandi(AT)tin.it), Feb 13 2009]
> %Y
> A078371 Cf. A046092, A141530 [From Vincenzo Librandi
> (vincenzo.librandi
> (AT)tin.it), Feb 13 2009]

Here, and in the two comments that follow, your "except" means that
you have to use these three sequences with different offsets.

I think I might say something more like this:
%C A078371 Define X(n) = A141530(n+2), A(n) = A078371(n), and Y(n) =
A046092(n+1).  Then X(n)^2 - A(n) * Y(n)^2 = 1.

But this looks like it might be a nice point about the polynomials
(especially since A141530 is otherwise a pretty uninteresting
sequence).

--Joshua Zucker

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