A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.
0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220, 27683, 39780, 55719, 79856, 114359, 163680, 234183, 327080, 467759, 668856, 956319, 1367240, 1908683, 2728620, 3900699, 5576156, 7971179, 11126940, 15905883, 22737260, 32502539, 46461756, 64854879, 92708600, 132524783
Offset: 1
Examples
For p=31 (m=4) the first five (5) consecutive solutions are (0, 961), (279, 1271), (656, 1745), (1139, 2389), (1860, 3379).
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6,0,0,0,-1,1).
Crossrefs
Programs
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Magma
I:=[0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+1922: n in [1..100]];
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Mathematica
LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220}, 36] (* Jean-François Alcover, Feb 12 2020 *) -
PARI
concat(0, Vec(x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^30))) \\ Colin Barker, Feb 12 2020
Formula
a(n) = 6*a(n-5) - a(n-10) + 1922 for n >= 11; a(1)=0, a(2)=279, a(3)=656, a(4)=1139, a(5)=1860, a(6)=2883, a(7)=4340, a(8)=6419, a(9)=9156, a(10)=13299.
From Colin Barker, Feb 12 2020: (Start)
G.f.: x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)
Comments