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

1,1

Comments

See A355881 for a general formula.

Links

Paolo Xausa, Table of n, a(n) for n = 1..800

Index entries for linear recurrences with constant coefficients, signature (18,-27).

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

Example

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

Mathematica

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

Crossrefs

Cf. A198710, A355881, A355883.

Sequence in context: A182279 A203760 A208935 * A202534 A277497 A173174

Adjacent sequences: A355879 A355880 A355881 * A355883 A355884 A355885

Keyword

nonn,easy

Author

Gerhard Kirchner, Jul 24 2022

Status

approved