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

Author

Klaus Brockhaus, Apr 11 2009

Keywords

• nonn
• easy

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.

Examples

			(-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.
		

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).

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).

Programs

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

Mathematica

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

PARI

{forstep(n=-60, 1100000000, [3,1], if(issquare(2*n^2+466*n+54289, &k),print1(k, ",")))};

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.