# [seqfan] Re: A1: In binary representation, a(n)^2 is the smallest square with n^2 as substring.

zak seidov zakseidov at yahoo.com
Tue Aug 18 18:26:24 CEST 2009

```Leroy, David, Richard,
thank you all for your great responces.

I'll copy them in my LJ.

Zak

--- On Tue, 8/18/09, Richard Mathar <mathar at strw.leidenuniv.nl> wrote:

> From: Richard Mathar <mathar at strw.leidenuniv.nl>
> Subject: [seqfan] Re: A1: In binary representation, a(n)^2  is the smallest square with n^2 as substring.
> To: seqfan at seqfan.eu
> Date: Tuesday, August 18, 2009, 12:10 PM
>
> as a comment on http://list.seqfan.eu/pipermail/seqfan/2009-August/002126.html
>
> dw> From seqfan-bounces at list.seqfan.eu
> Tue Aug 18 17:11:09 2009
> dw> Date: Tue, 18 Aug 2009 10:38:44 -0400
> dw> From: David Wilson <dwilson at gambitcomm.com>
> dw> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> dw> Subject: [seqfan] Re: A1: In binary representation,
> a(n)^2  is the smallest square with n^2 as substring.
> dw> ...
> dw> The only nontrivial part is solving k^2 = 2n^2+1. To
> dw>
> dw>     (n, k) = (0, 1)
> dw>
> dw> and repeatedly apply the map
> dw>
> dw>     (n, k) => (3n+2k, 4n+3k)
> dw>
> dw> This produces the pairs
> dw>
> dw>     (0, 1) => (2, 3) => (12,
> 17) => (70, 99) => ...
> dw>
> dw> so that
> dw>
> dw>     f(0) = 1; f(2) = 3; f(12) = 17;
> f(70) = 99; etc.
>
> the lazy OEIS user might observe that this generates
> A001542 for n and A001541 for k.
>
> RJM
>
>
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>

```