A160055 Positive numbers y such that y^2 is of the form x^2+(x+89)^2 with integer x.

65, 89, 149, 241, 445, 829, 1381, 2581, 4825, 8045, 15041, 28121, 46889, 87665, 163901, 273289, 510949, 955285, 1592845, 2978029, 5567809, 9283781, 17357225, 32451569, 54109841, 101165321, 189141605, 315375265, 589634701, 1102398061

Offset: 1

Author

Klaus Brockhaus, May 04 2009

Keywords

• nonn

Comments

(-33, a(1)) and (A129298(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 1.

Examples

			(-33, a(1)) = (-33, 65) is a solution: (-33)^2+(-33+89)^2 = 1089+3136 = 4225 = 65^2.
(A129298(1), a(2)) = (0, 89) is a solution: 0^2+(0+89)^2 = 7921 = 89^2.
(A129298(3), a(4)) = (120, 241) is a solution: 120^2+(120+89)^2 = 14400+43681 = 58081 = 241^2.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129298, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{65,89,149,241,445,829},40] (* Harvey P. Dale, Feb 04 2015 *)

PARI

{forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=65, a(2)=89, a(3)=149, a(4)=241, a(5)=445, a(6)=829.

G.f.: (1-x)*(65+154*x+303*x^2+154*x^3+65*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 89*A001653(k) for k >= 1.