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

1,1

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.

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).

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.

Example

(-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.

Mathematica

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

Prog

(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
(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

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).

Sequence in context: A048055 A067803 A098258 * A077085 A165989 A183598

Adjacent sequences: A160173 A160174 A160175 * A160177 A160178 A160179

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved