%I #14 Feb 15 2020 10:52:27
%S 0,200,611,1227,2291,4620,8180,14364,27927,48671,84711,163760,284664,
%T 494720,955451,1660131,2884427,5569764,9676940,16812660,32463951,
%U 56402327,97992351,189214760,328737840,571142264,1102825427,1916025531,3328862051,6427738620
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+409, y).
%C Corresponding values y of solutions (x, y) are in A160577.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.
%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)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.
%F G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 409*A001652(k) for k >= 0.
%t LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)
%o (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ",")))}
%Y Cf. A160577, A001652, A129640, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, May 31 2007
%E Edited and two terms added by _Klaus Brockhaus_, Jun 08 2009