A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

0, 111, 1260, 1707, 2280, 8791, 11380, 14707, 52624, 67711, 87100, 308091, 396024, 509031, 1797060, 2309571, 2968224, 10475407, 13462540, 17301451, 61056520, 78466807, 100841620, 355864851, 457339440, 587749407, 2074133724, 2665570971

Offset: 1

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+569, y).

Corresponding values y of solutions (x, y) are in A160090.

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

lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 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. A160090, A129298, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).

Programs

Magma

I:=[0,111,1260,1707,2280,8791,11380]; [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, Apr 21 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,111,1260,1707,2280,8791,11380}, 50] (* G. C. Greubel, Apr 21 2018 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761), print1(n, ",")))}

PARI

x='x+O('x^30); concat([0], Vec(x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, Apr 21 2018

Formula

a(n) = 6*a(n-3) - a(n-6) + 1138 for n > 6; a(1)=0, a(2)=111, a(3)=1260, a(4)=1707, a(5)=2280, a(6)=8791.

G.f.: x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)).

a(3*k+1) = 569*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 04 2009