A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

0, 132, 2295, 2859, 3535, 15792, 19060, 22984, 94363, 113407, 136275, 552292, 663288, 796572, 3221295, 3868227, 4645063, 18777384, 22547980, 27075712, 109444915, 131421559, 157811115, 637894012, 765983280, 919792884, 3717921063

Offset: 1

Author

Mohamed Bouhamida, Jun 13 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+953, y).

Corresponding values y of solutions (x, y) are in A160212.

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

lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 0.

Links

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

Crossrefs

Cf. A160212, A001652, A129974, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,132,2295,2859,3535,15792,19060},30] (* Harvey P. Dale, Apr 12 2013 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1906 for n > 6; a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792.

G.f.: x*(132+2163*x+564*x^2-116*x^3-721*x^4-116*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 953*A001652(k) for k >= 0.

a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792, a(7)=19060, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Apr 12 2013

Extensions

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