A161478 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

0, 52, 175, 339, 615, 1312, 2260, 3864, 7923, 13447, 22795, 46452, 78648, 133132, 271015, 458667, 776223, 1579864, 2673580, 4524432, 9208395, 15583039, 26370595, 53670732, 90824880, 153699364, 312816223, 529366467, 895825815, 1823226832, 3085374148

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

Corresponding values y of solutions (x, y) are in A161479.

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

lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A161479, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,52,175,339,615,1312,2260},72] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

PARI

{forstep(n=0, 100000000, [3, 1], if(issquare(2*n^2+226*n+12769), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+226 for n > 6; a(1)=0, a(2)=52, a(3)=175, a(4)=339, a(5)=615, a(6)=1312.

G.f.: x*(52+123*x+164*x^2-36*x^3-41*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 113*A001652(k) for k >= 0.