A130013 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

0, 37, 1768, 1941, 2128, 11937, 12940, 14025, 71148, 76993, 83316, 416245, 450312, 487165, 2427616, 2626173, 2840968, 14150745, 15308020, 16559937, 82478148, 89223241, 96519948, 480719437, 520032720, 562561045, 2801839768, 3030974373

Offset: 1

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+647, y).

Corresponding values y of solutions (x, y) are in A159641.

For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.

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

lim_{n -> infinity} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 0.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A159641, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,37,1768,1941,2128,11937,12940},40] (* Harvey P. Dale, Jan 27 2025 *)

PARI

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.

G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 647*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 21 2009