A129298 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.

0, 51, 120, 267, 540, 931, 1780, 3367, 5644, 10591, 19840, 33111, 61944, 115851, 193200, 361251, 675444, 1126267, 2105740, 3936991, 6564580, 12273367, 22946680, 38261391, 71534640, 133743267, 223003944, 416934651, 779513100, 1299762451

Offset: 1

Author

Mohamed Bouhamida, May 26 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+89, y).

Corresponding values y of solutions (x, y) are in A160055.

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

lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 0.

Links

• G. C. Greubel, 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. A160055, A129289, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Programs

Magma

I:=[0,51,120,267,540,931,1780]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,51,120,267,540,931,1780},30] (* Harvey P. Dale, Sep 21 2013 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921), print1(n, ",")))};

PARI

x='x+O('x^30); concat(0, Vec(x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, Apr 19 2018

Formula

a(n) = 6*a(n-3) - a(n-6) + 178 with for n > 6; a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931.

G.f.: x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 89*A001652(k), k >= 0. (Zak Seidov, May 28 2007)

a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931, a(7)=1780, a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7). - Harvey P. Dale, Sep 21 2013

Extensions

Edited and three terms added by Klaus Brockhaus, May 04 2009