A159466 Positive numbers y such that y^2 is of the form x^2 + (x+127)^2 with integer x.

113, 127, 145, 533, 635, 757, 3085, 3683, 4397, 17977, 21463, 25625, 104777, 125095, 149353, 610685, 729107, 870493, 3559333, 4249547, 5073605, 20745313, 24768175, 29571137, 120912545, 144359503, 172353217, 704729957, 841388843, 1004548165

Offset: 1

Author

Klaus Brockhaus, Apr 13 2009

Keywords

• nonn
• easy

Comments

(-15, a(1)) and (A129992(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.

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

Lim_{n -> infinity} a(n)/a(n-1) = (129 + 16*sqrt(2))/127 for n mod 3 = {0, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (34947 + 21922*sqrt(2))/127^2 for n mod 3 = 1.

Examples

			(-15, a(1)) = (-15, 113) is a solution: (-15)^2 + (-15+127)^2 = 225 + 12544 = 12769 = 113^2.
(A129992(1), a(2)) = (0, 127) is a solution: 0^2 + (0+127)^2 = 16129 = 127^2.
(A129992(3), a(4)) = (308, 533) is a solution: 308^2 + (308+127)^2 = 94864 + 189225 = 284089 = 533^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..3900
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129992, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).

Programs

Magma

I:=[113,127,145,533,635,757]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 15 2018

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{113,127,145,533,635,757},50] (* Harvey P. Dale, Feb 06 2015 *)

PARI

{forstep(n=-16, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3) - a(n-6)for n > 6; a(1)=113, a(2)=127, a(3)=145, a(4)=533, a(5)=635, a(6)=757.

G.f.: (1-x)*(113+240*x+385*x^2+240*x^3+113*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 127*A001653(k) for k >= 1.