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A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

%I A101152 #16 Sep 08 2022 08:45:16
%S A101152 0,111,1260,1707,2280,8791,11380,14707,52624,67711,87100,308091,
%T A101152 396024,509031,1797060,2309571,2968224,10475407,13462540,17301451,
%U A101152 61056520,78466807,100841620,355864851,457339440,587749407,2074133724,2665570971
%N A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.
%C A101152 Also values x of Pythagorean triples (x, x+569, y).
%C A101152 Corresponding values y of solutions (x, y) are in A160090.
%C A101152 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A101152 lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {1, 2}.
%C A101152 lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 for n mod 3 = 0.
%H A101152 G. C. Greubel, <a href="/A101152/b101152.txt">Table of n, a(n) for n = 1..1000</a>
%H A101152 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A101152 a(n) = 6*a(n-3) - a(n-6) + 1138 for n > 6; a(1)=0, a(2)=111, a(3)=1260, a(4)=1707, a(5)=2280, a(6)=8791.
%F A101152 G.f.: x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)).
%F A101152 a(3*k+1) = 569*A001652(k) for k >= 0.
%t A101152 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,111,1260,1707,2280,8791,11380}, 50] (* _G. C. Greubel_, Apr 21 2018 *)
%o A101152 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761), print1(n, ",")))}
%o A101152 (PARI) x='x+O('x^30); concat([0], Vec(x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ _G. C. Greubel_, Apr 21 2018
%o A101152 (Magma) I:=[0,111,1260,1707,2280,8791,11380]; [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_, Apr 21 2018
%Y A101152 Cf. A160090, A129298, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).
%K A101152 nonn,easy
%O A101152 1,2
%A A101152 _Mohamed Bouhamida_, Jun 03 2007
%E A101152 Edited and two terms added by _Klaus Brockhaus_, May 04 2009