A332000 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 47^2)^2 = y^2.
0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180, 67116, 101664, 115227, 174135, 262871, 396539, 597891, 676940, 1020276, 1537464, 2316536, 3490100, 3950831, 5951939, 8966331, 13507095, 20347127, 23032464, 34695776, 52264940, 78730452, 118597080, 134248371
Offset: 1
Examples
For p=47 (m=7) the first five (5) consecutive solutions are (0, 2209), (752, 3055), (1820, 4421), (2231, 4969), (3995, 7379).
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, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+4418: n in [1..100]];
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Mathematica
LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180}, 40] (* Jean-François Alcover, Feb 08 2020 *) -
PARI
concat(0, Vec(x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^40))) \\ Colin Barker, Feb 04 2020
Formula
a(n) = 6*a(n-5) - a(n-10) + 4418 for n >= 11; a(1)=0, a(2)=752, a(3)=1820, a(4)=2231, a(5)=3995, a(6)=6627, a(7)=10575, a(8)=16511, a(9)=18840, a(10)=28952.
From Colin Barker, Feb 04 2020: (Start)
G.f.: x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*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