cp's OEIS Frontend

A129640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

%I A129640 #14 Feb 15 2020 10:52:27
%S A129640 0,155,464,939,1764,3515,6260,11055,21252,37247,65192,124623,217848,
%T A129640 380723,727112,1270467,2219772,4238675,7405580,12938535,24705564,
%U A129640 43163639,75412064,143995335,251576880,439534475,839267072,1466298267,2561795412,4891607723
%N A129640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.
%C A129640 Also values x of Pythagorean triples (x, x+313, y).
%C A129640 Corresponding values y of solutions (x, y) are in A160574.
%C A129640 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129640 lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {1, 2}.
%C A129640 lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 0.
%H A129640 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129640 a(n) = 6*a(n-3)-a(n-6)+626 for n > 6; a(1)=0, a(2)=155, a(3)=464, a(4)=939, a(5)=1764, a(6)=3515.
%F A129640 G.f.: x*(155+309*x+475*x^2-105*x^3-103*x^4-105*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A129640 a(3*k+1) = 313*A001652(k) for k >= 0.
%t A129640 LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 155, 464, 939, 1764, 3515, 6260}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)
%o A129640 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969), print1(n, ",")))}
%Y A129640 Cf. A160574, A001652, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).
%K A129640 nonn,easy
%O A129640 1,2
%A A129640 _Mohamed Bouhamida_, May 31 2007
%E A129640 Edited and two terms added by _Klaus Brockhaus_, Jun 08 2009