A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708

Offset: 1

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+223, y).

Corresponding values y of solutions (x, y) are in A159809.

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) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,32,533,669,833,3672,4460}, 70]  (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

PARI

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.

G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 223*A001652(k) for k >= 0.

Extensions

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