A129725 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

0, 100, 1159, 1563, 2079, 8080, 10420, 13416, 48363, 61999, 79459, 283140, 362616, 464380, 1651519, 2114739, 2707863, 9627016, 12326860, 15783840, 56111619, 71847463, 91996219, 327043740, 418758960, 536194516, 1906151863, 2440707339, 3125171919, 11109868480

Offset: 1

Author

Mohamed Bouhamida, Jun 02 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+521, y).

Corresponding values y of solutions (x, y) are in A160583.

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

lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160583, A001652, A129642, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

Programs

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 100, 1159, 1563, 2079, 8080, 10420}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1042 for n > 6; a(1)=0, a(2)=100, a(3)=1159, a(4)=1563, a(5)=2079, a(6)=8080.

G.f.: x*(100+1059*x+404*x^2-84*x^3-353*x^4-84*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 521*A001652(k) for k >= 0.

Extensions

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