A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

0, 76, 559, 843, 1239, 3976, 5620, 7920, 23859, 33439, 46843, 139740, 195576, 273700, 815143, 1140579, 1595919, 4751680, 6648460, 9302376, 27695499, 38750743, 54218899, 161421876, 225856560, 316011580, 940836319, 1316389179, 1841851143, 5483596600

Offset: 1

Author

Mohamed Bouhamida, May 30 2007

Keywords

• nonn
• easy

Comments

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

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

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

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

lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

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).

Programs

Mathematica

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

PARI

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

Formula

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.

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)).

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

Extensions

Edited by Klaus Brockhaus, Apr 12 2009