A355882 Number of ways to 4-color a 3 X n grid ignoring the variations of two colors.

3, 49, 801, 13095, 214083, 3499929, 57218481, 935434575, 15292923363, 250015887009, 4087377035361, 66822357687255, 1092443258415843, 17859774993929289, 291979981913499441, 4773425749606899135, 78038203981259699523, 1275805176423288314769

Offset: 1

Author

Gerhard Kirchner, Jul 24 2022

Keywords

• nonn
• easy

Comments

See A355881 for a general formula.

Examples

			a(1) = 3, 4 colors 1,2,3,4: 121, 123, 124.
The first two colors do not vary.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..800
• Index entries for linear recurrences with constant coefficients, signature (18,-27).

Crossrefs

Cf. A198710, A355881, A355883.

Programs

Mathematica

LinearRecurrence[{18, -27}, {3, 49}, 20] (* Paolo Xausa, Oct 03 2024 *)

Formula

G.f.: x*(3-5*x)/(1-18*x+27*x^2).

a(n) = 18*a(n-1) - 27*a(n-2) with a(1) = 3, a(2) = 49.

a(n) = 3^(n-7/2)*((12 + 5*sqrt(6))*(3 + sqrt(6))^n - (3 - sqrt(6))^n*(12 - 5*sqrt(6)))/(2*sqrt(2)). - Stefano Spezia, Jul 24 2022

a(n) = 2*A198710(n) - 1. - Hugo Pfoertner, Jul 24 2022