A342710 Solutions x to the Pell-Fermat equation x^2 - 5*y^2 = 4.

3, 18, 123, 843, 5778, 39603, 271443, 1860498, 12752043, 87403803, 599074578, 4106118243, 28143753123, 192900153618, 1322157322203, 9062201101803, 62113250390418, 425730551631123, 2918000611027443, 20000273725560978, 137083915467899403, 939587134549734843

Offset

0,1

Comments

This Pell equation is used to find the 12-gonal square numbers (see A342709).

The corresponding solutions y are in A033890.

Essentially the same as A246453. - R. J. Mathar, Mar 24 2021

Links

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (7,-1).

Formula

a(n) = 7*a(n-1) - a(n-2).

a(n) = 2*T(2*n+1, 3/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 02 2022

Example

a(1)^2 - 5 * A033890(1)^2 = 18^2 - 5 * 8^2 = 4.

Mathematica

LinearRecurrence[{7, -1}, {3, 18}, 20] (* Amiram Eldar, Mar 19 2021 *)

Crossrefs

Cf. A033890, A342709.

a(n) = 3*A049685(n). - Hugo Pfoertner, Mar 19 2021

Sequence in context: A074558 A074564 A108241 * A199421 A305869 A371483

Adjacent sequences: A342707 A342708 A342709 * A342711 A342712 A342713

Keyword

nonn,easy

Author

Bernard Schott, Mar 19 2021

Status

approved