A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

0, 200, 611, 1227, 2291, 4620, 8180, 14364, 27927, 48671, 84711, 163760, 284664, 494720, 955451, 1660131, 2884427, 5569764, 9676940, 16812660, 32463951, 56402327, 97992351, 189214760, 328737840, 571142264, 1102825427, 1916025531, 3328862051, 6427738620

Offset: 1

Author

Mohamed Bouhamida, May 31 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+409, y).

Corresponding values y of solutions (x, y) are in A160577.

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

lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160577, A001652, A129640, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

Programs

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.

G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 409*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Jun 08 2009