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

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset: 1

Author

Mohamed Bouhamida, May 30 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+137, y).

Corresponding values y of solutions (x, y) are in A157213.

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

lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.

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.

Links

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

Crossrefs

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.

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

PARI

{forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}

Formula

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.

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)).

a(3*k+1) = 137*A001652(k) for k >= 0.

Extensions

Edited and extended by Klaus Brockhaus, Feb 25 2009