A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

0, 75, 432, 699, 1092, 3115, 4660, 6943, 18724, 27727, 41032, 109695, 162168, 239715, 639912, 945747, 1397724, 3730243, 5512780, 8147095, 21742012, 32131399, 47485312, 126722295, 187276080, 276765243, 738592224, 1091525547, 1613106612

Offset: 1

Author

Mohamed Bouhamida, May 30 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+233, y).

Corresponding values y of solutions (x, y) are in A157297.

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 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 0.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A157297, A001652, A129288, A129289, A129298, 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:=[0,75,432,699,1092,3115,4660]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..30]]; // G. C. Greubel, Mar 29 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* G. C. Greubel, Mar 29 2018 *)

PARI

{forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.

G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).

a(3*k+1) = 233*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 11 2009