A129626 - OEIS

%I #14 Feb 15 2020 10:52:27

%S 0,76,559,843,1239,3976,5620,7920,23859,33439,46843,139740,195576,

%T 273700,815143,1140579,1595919,4751680,6648460,9302376,27695499,

%U 38750743,54218899,161421876,225856560,316011580,940836319,1316389179,1841851143,5483596600

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+281, y).

%C Corresponding values y of solutions (x, y) are in A157348.

%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 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)+562 for n > 6; a(1)=0, a(2)=76, a(3)=559, a(4)=843, a(5)=1239, a(6)=3976.

%F G.f.: x*(76+483*x+284*x^2-60*x^3-161*x^4-60*x^5)/((1-x)*(1-6*x^3+x^6)).

%F a(3*k+1) = 281*A001652(k) for k >= 0.

%t LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 76, 559, 843, 1239, 3976, 5620}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)

%o (PARI) {forstep(n=0, 1000000000, [3, 1], if(issquare(2*n^2+562*n+78961), print1(n, ",")))}

%Y Cf. A157348, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

%K nonn,easy

%O 1,2

%A _Mohamed Bouhamida_, May 30 2007

%E Edited by _Klaus Brockhaus_, Apr 12 2009