A202299 y-values in the solution to x^2 - 18*y^2 = 1.

0, 4, 136, 4620, 156944, 5331476, 181113240, 6152518684, 209004522016, 7100001229860, 241191037293224, 8193395266739756, 278334248031858480, 9455171037816448564, 321197481037727392696, 10911259184244914903100

Offset: 1

Author

Vincenzo Librandi, Dec 18 2011

Keywords

• nonn
• easy

Comments

The corresponding values of x of this Pell equation are in A056771.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Index entries for linear recurrences with constant coefficients, signature (34,-1).

Crossrefs

Cf. A001542, A056771.

Programs

Magma

I:=[0, 4]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]];

Mathematica

LinearRecurrence[{34,-1},{0,4},30]
With[{c=6*Sqrt[2]},Table[((17-2c)^n-(17+2c)^n)/-c,{n,0,20}]]//Simplify (* Harvey P. Dale, Dec 16 2024 *)

Maxima

makelist(expand(((3+2*sqrt(2))^(2*n-2)-(3-2*sqrt(2))^(2*n-2))/(6*sqrt(2))), n, 1, 16); /* _Bruno Berselli, Dec 19 2011 */

Formula

a(n) = 34*a(n-1)-a(n-2) with a(1)=0, a(2)=4.

G.f.: 4*x^2/(1-34*x+x^2).

a(n) = (1/3)*A001542(2n-2). - Bruno Berselli, Dec 19 2011