A129974 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

0, 627, 1128, 2811, 6188, 9027, 18740, 38375, 54908, 111503, 225936, 322295, 652152, 1319115, 1880736, 3803283, 7690628, 10963995, 22169420, 44826527, 63905108, 129215111, 261270408, 372468527, 753123120, 1522797795, 2170907928

Offset: 1

Author

Mohamed Bouhamida, Jun 13 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+937, y).

Corresponding values y of solutions (x, y) are in A160209.

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

lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160209, A001652, A129857, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).

Programs

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1874 for n > 6; a(1)=0, a(2)=627, a(3)=1128, a(4)=2811, a(5)=6188, a(6)=9027.

G.f.: x*(627+501*x+1683*x^2-385*x^3-167*x^4-385*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 937*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009