A160176 Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x.

533, 617, 733, 2465, 3085, 3865, 14257, 17893, 22457, 83077, 104273, 130877, 484205, 607745, 762805, 2822153, 3542197, 4445953, 16448713, 20645437, 25912913, 95870125, 120330425, 151031525, 558772037, 701337113, 880276237

Offset: 1

Author

Klaus Brockhaus, May 18 2009

Keywords

• nonn

Comments

(-92, a(1)) and (A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.

Examples

			(-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
(A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2.
(A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).

Programs

Magma

I:=[533,617,733,2465,3085,3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 04 2018

Mathematica

LinearRecurrence[{0,0,6,0,0,-1}, {533,617,733,2465,3085,3865}, 50] (* G. C. Greubel, May 04 2018 *)

PARI

{forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ",")))}

PARI

x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, May 04 2018

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.

G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 617*A001653(k) for k >= 1.