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%I #17 Oct 21 2022 22:09:18
%S 0,51,120,267,540,931,1780,3367,5644,10591,19840,33111,61944,115851,
%T 193200,361251,675444,1126267,2105740,3936991,6564580,12273367,
%U 22946680,38261391,71534640,133743267,223003944,416934651,779513100,1299762451
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+89, y).
%C Corresponding values y of solutions (x, y) are in A160055.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 0.
%H G. C. Greubel, <a href="/A129298/b129298.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F a(n) = 6*a(n-3) - a(n-6) + 178 with for n > 6; a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931.
%F G.f.: x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 89*A001652(k), k >= 0. (Zak Seidov, May 28, 2007)
%F a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931, a(7)=1780, a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7). - _Harvey P. Dale_, Sep 21 2013
%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,51,120,267,540,931,1780},30] (* _Harvey P. Dale_, Sep 21 2013 *)
%o (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921), print1(n, ",")))};
%o (PARI) x='x+O('x^30); concat(0, Vec(x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ _G. C. Greubel_, Apr 19 2018
%o (Magma) I:=[0,51,120,267,540,931,1780]; [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 19 2018
%Y Cf. A160055, A129289, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, May 26 2007
%E Edited and three terms added by _Klaus Brockhaus_, May 04 2009
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