A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.

185, 233, 317, 793, 1165, 1717, 4573, 6757, 9985, 26645, 39377, 58193, 155297, 229505, 339173, 905137, 1337653, 1976845, 5275525, 7796413, 11521897, 30748013, 45440825, 67154537, 179212553, 264848537, 391405325, 1044527305, 1543650397

Offset

1,1

Comments

(-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1.

Links

G. C. Greubel, 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)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.

G.f.: (1-x)*(185 +418*x +735*x^2 +418*x^3 +185*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 233*A001653(k) for k >= 1.

Example

(-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.
(A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.
(A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {185, 233, 317, 793, 1165, 1717}, 50] (* G. C. Greubel, Mar 29 2018 *)

Prog

(PARI) {forstep(n=-60, 1100000000, [3, 1], if(issquare(2*n^2+466*n+54289, &k), print1(k, ", ")))};
(Magma) I:=[185, 233, 317, 793, 1165, 1717]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Mar 29 2018

Crossrefs

Cf. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

Sequence in context: A353388 A262053 A362096 * A156059 A244248 A129311

Adjacent sequences: A157294 A157295 A157296 * A157298 A157299 A157300

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 11 2009

Status

approved