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

0, 155, 464, 939, 1764, 3515, 6260, 11055, 21252, 37247, 65192, 124623, 217848, 380723, 727112, 1270467, 2219772, 4238675, 7405580, 12938535, 24705564, 43163639, 75412064, 143995335, 251576880, 439534475, 839267072, 1466298267, 2561795412, 4891607723

Offset: 1

Author

Mohamed Bouhamida, May 31 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+313, y).

Corresponding values y of solutions (x, y) are in A160574.

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

lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. 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).

Programs

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 155, 464, 939, 1764, 3515, 6260}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969), print1(n, ",")))}

Formula

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.

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)).

a(3*k+1) = 313*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Jun 08 2009