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A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

%I A129544 #13 Feb 15 2020 10:52:27
%S A129544 0,115,136,411,1036,1155,2740,6375,7068,16303,37488,41527,95352,
%T A129544 218827,242368,556083,1275748,1412955,3241420,7435935,8235636,
%U A129544 18892711,43340136,48001135,110115120,252605155,279771448,641798283,1472291068,1630627827,3740674852
%N A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.
%C A129544 Also values x of Pythagorean triples (x, x+137, y).
%C A129544 Corresponding values y of solutions (x, y) are in A157213.
%C A129544 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129544 lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.
%C A129544 lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 0.
%H A129544 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129544 a(n) = 6*a(n-3)-a(n-6)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.
%F A129544 G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A129544 a(3*k+1) = 137*A001652(k) for k >= 0.
%t A129544 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)
%o A129544 (PARI) {forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}
%Y A129544 Cf. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.
%K A129544 nonn,easy
%O A129544 1,2
%A A129544 _Mohamed Bouhamida_, May 30 2007
%E A129544 Edited and extended by _Klaus Brockhaus_, Feb 25 2009