A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

0, 28, 385, 501, 645, 2668, 3340, 4176, 15957, 19873, 24745, 93408, 116232, 144628, 544825, 677853, 843357, 3175876, 3951220, 4915848, 18510765, 23029801, 28652065, 107889048, 134227920, 166996876, 628823857, 782338053, 973329525, 3665054428, 4559800732

Offset: 1

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+167, y).

Corresponding values y of solutions (x, y) are in A159777.

For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.

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

lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 0.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A159777, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,28,385,501,645,2668,3340},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

PARI

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+334*n+27889), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+334 for n > 6; a(1)=0, a(2)=28, a(3)=385, a(4)=501, a(5)=645, a(6)=2668.

G.f.: x*(28+357*x+116*x^2-24*x^3-119*x^4-24*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 167*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 30 2009