A160098 Positive numbers y such that y^2 is of the form x^2+(x+601)^2 with integer x.

425, 601, 1261, 1289, 3005, 7141, 7309, 17429, 41585, 42565, 101569, 242369, 248081, 591985, 1412629, 1445921, 3450341, 8233405, 8427445, 20110061, 47987801, 49118749, 117210025, 279693401, 286285049, 683150089, 1630172605

Offset

1,1

Comments

(-297, a(1)) and (A111258(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 1.

Links

G. C. Greubel, Table of n, a(n) for n = 1..3875

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)=425, a(2)=601, a(3)=1261, a(4)=1289, a(5)=3005, a(6)=7141.

G.f.: (1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 601*A001653(k) for k >= 1.

Example

(-297, a(1)) = (-297, 425) is a solution: (-297)^2+(-297+601)^2 = 88209+92416 = 180625 = 425^2.
(A111258(1), a(2)) = (0, 601) is a solution: 0^2+(0+601)^2 = 361201 = 601^2.
(A111258(3), a(4)) = (560, 1289) is a solution: 560^2+(560+601)^2 = 313600+1347921 = 1661521 = 1289^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {425, 601, 1261, 1289, 3005, 7141}, 50] (* G. C. Greubel, Apr 22 2018 *)

Prog

(PARI) {forstep(n=-300, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 22 2018
(Magma) I:=[425, 601, 1261, 1289, 3005, 7141]; [n le 6 select I[n] else 5*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 22 2018

Crossrefs

Cf. A111258, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).

Sequence in context: A232359 A294714 A294712 * A349940 A203343 A207233

Adjacent sequences: A160095 A160096 A160097 * A160099 A160100 A160101

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved