[seqfan] Re: A sequence connected to the Pythagorean triples

[Tom Duff]

It's not clear which is better. The triangle makes it annoying to extract
the sequence in the "increasing b within increasing c" order if that's what
you need to find and vice versa. The usual solution is to submit both.

On Tue, Jun 14, 2022 at 3:30 AM Yifan Xie <xieyifan4013 at 163.com> wrote:

> Hi,
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>
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>
> I think it's better to write the terms in a triangle T(m, k) satisfying
> m>0, k>0, k<m.
>
>
> The triangle begins:
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>  1,
>
>  4,  5
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> 10, 16, 14
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> 20, 35, 40, 30
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> 35, 64, 81, 80, 55
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> ...
>
> Is this clearer?
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> I also checked the row sums, which begins in
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> 1, 9, 40, 125, 315,
>
> is A006414.
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> I couldn't find the term 9.
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> Best regards,
>
>
>
>
> Yifan Xie
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> At 2022-06-12 15:06:12, "Ali Sada via SeqFan" <seqfan at list.seqfan.eu>
> wrote:
> >Hi everyone,
> >
> >Just wanted to check if this sequence is suitable for the OEIS.
> >
> >a(n) = ((c^4-a^4-b^4)/288)^1/2, where a, b, and c are the Pythagorean
> triples ordered by increasing c.
> >
> >Let S be c^4-a^4-b^4
> >Using the Pythagorean triples generators
> >a = 2mk
> >b = m^2-k^2
> >c = m^2+k^2
> >Then we substitute to get the value of S in terms of m and k
> >S = 8*m^2*k^2*((m^2-k^2))^2
> >That’s a multiple of 288. When we divide by 288, we get
> >
> >1, 16, 25, 81, 100, 256, 625, 196, 400, 1225, 1296, 1600, 2401, 1225,
> 2025, 4096, 900, 6561, 10000, 3136, 8100, 6400, 11025, 14641, 19600, 20736,
> 3025, 28561, 23716, 15625, 7056, 25600, 38416, 48400, 19600, 50625, 15876,
> 32400, 65536, 14400, 83521, 62500, 53361, 8281, 99225, 67600, 104976,
> 60025, 130321, 152100, 160000, 50176, 27225, 129600, 102400, 194481,
> 176400, 207025, 234256, 99225, 19600, 279841, 313600, 164025, 240100, 331776
> >
> >We take the square roots, and we get a(n)
> >
> >1, 4, 5, 9, 10, 16, 25, 14, 20, 35, 36, 40, 49, 35, 45, 64, 30, 81, 100,
> 56, 90, 80, 105, 121, 140, 144, 55, 169, 154, 125, 84, 160, 196, 220, 140,
> 225, 126, 180, 256, 120, 289, 250, 231, 91, 315, 260, 324, 245, 361, 390,
> 400, 224, 165, 360, 320, 441, 420, 455, 484, 315, 140, 529, 560, 405, 490,
> 576>Best,
> >Ali>
> >--
> >Seqfan Mailing list - http://list.seqfan.eu/
>
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>