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A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

%I A129625 #13 Sep 08 2022 08:45:30
%S A129625 0,75,432,699,1092,3115,4660,6943,18724,27727,41032,109695,162168,
%T A129625 239715,639912,945747,1397724,3730243,5512780,8147095,21742012,
%U A129625 32131399,47485312,126722295,187276080,276765243,738592224,1091525547,1613106612
%N A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.
%C A129625 Also values x of Pythagorean triples (x, x+233, y).
%C A129625 Corresponding values y of solutions (x, y) are in A157297.
%C A129625 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129625 lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {1, 2}.
%C A129625 lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 0.
%H A129625 G. C. Greubel, <a href="/A129625/b129625.txt">Table of n, a(n) for n = 1..1000</a>
%H A129625 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129625 a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.
%F A129625 G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).
%F A129625 a(3*k+1) = 233*A001652(k) for k >= 0.
%t A129625 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* _G. C. Greubel_, Mar 29 2018 *)
%o A129625 (PARI) {forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};
%o A129625 (Magma) I:=[0,75,432,699,1092,3115,4660]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..30]]; // _G. C. Greubel_, Mar 29 2018
%Y A129625 Cf. A157297, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).
%K A129625 nonn,easy
%O A129625 1,2
%A A129625 _Mohamed Bouhamida_, May 30 2007
%E A129625 Edited and two terms added by _Klaus Brockhaus_, Apr 11 2009