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

1,1

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.

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).

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.

Example

(-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.

Mathematica

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

Prog

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

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).

Sequence in context: A015788 A072053 A336331 * A250642 A280755 A020140

Adjacent sequences: A160052 A160053 A160054 * A160056 A160057 A160058

Keyword

nonn

Author

Klaus Brockhaus, May 04 2009

Status

approved