A161479 Positive numbers y such that y^2 is of the form x^2+(x+113)^2 with integer x.

85, 113, 173, 337, 565, 953, 1937, 3277, 5545, 11285, 19097, 32317, 65773, 111305, 188357, 383353, 648733, 1097825, 2234345, 3781093, 6398593, 13022717, 22037825, 37293733, 75901957, 128445857, 217363805, 442389025, 748637317, 1266889097

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

(-36, a(1)) and (A161478(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

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 = {0, 2}.

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

Examples

			(-36, a(1)) = (-36, 85) is a solution: (-36)^2+(-36+113)^2 = 1296+5929 = 7225 = 85^2.
(A161478(1), a(2)) = (0, 113) is a solution: 0^2+(0+113)^2 = 12769 = 113^2.
(A161478(3), a(4)) = (175, 337) is a solution: 175^2+(175+113)^2 = 30625+82944 = 113569 = 337^2.
		

Crossrefs

Cf. A161478, A001653, 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

PARI

{forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+226*n+12769, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=85, a(2)=113, a(3)=173, a(4)=337, a(5)=565, a(6)=953.

G.f.: (1-x)*(85+198*x+371*x^2+198*x^3+85*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 113*A001653(k) for k >= 1.