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

Author

Bernard Schott, Mar 19 2021

Keywords

• nonn
• easy
• changed

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

Examples

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

Links

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

Crossrefs

Cf. A033890, A342709.

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

Programs

Mathematica

LinearRecurrence[{7, -1}, {3, 18}, 20] (* Amiram Eldar, Mar 19 2021 *)
Table[2 ChebyshevT[2 n + 1, 3/2], {n, 0, 20}] (* Eric W. Weisstein, Sep 02 2025 *)
Table[2 Cos[(2 n + 1) ArcCos[3/2]], {n, 0, 20}] // FunctionExpand (* Eric W. Weisstein, Sep 02 2025 *)

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

From Stefano Spezia, Apr 14 2025: (Start)

G.f.: 3*(1 - x)/(1 - 7*x + x^2).

E.g.f.: exp(7*x/2)*(3*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2)). (End)

a(n) = 2*cos((2*n+1)*arccos(3/2)). - Eric W. Weisstein, Sep 02 2025