A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

0, 56, 1925, 2181, 2465, 13056, 14540, 16188, 77865, 86513, 96117, 455588, 505992, 561968, 2657117, 2950893, 3277145, 15488568, 17200820, 19102356, 90275745, 100255481, 111338445, 526167356, 584333520, 648929768, 3066729845

Offset: 1

Author

Mohamed Bouhamida, Jun 20 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+727, y).

Corresponding values y of solutions (x, y) are in A159893.

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) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* Harvey P. Dale, Jan 16 2013 *)

PARI

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.

G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 727*A001652(k) for k >= 0.

Extensions

Edited and one term added by Klaus Brockhaus, Apr 30 2009