A320947 a(n) is the number of dominoes, among all domino tilings of the 2 X n rectangle, sharing a length-2 side with the boundary of the rectangle.

1, 4, 8, 16, 30, 56, 102, 184, 328, 580, 1018, 1776, 3082, 5324, 9160, 15704, 26838, 45736, 77742, 131840, 223112, 376844, 635378, 1069536, 1797650, 3017236, 5057672, 8467744, 14161038, 23657240, 39482358, 65832136, 109671112, 182552404, 303629290

Offset: 1

Author

Bridget Tenner, Oct 24 2018

Keywords

• nonn
• easy

Comments

a(n) is also the number of dominoes, among all domino tilings of the 2 x n rectangle, sharing a contiguous path of length at least 2 with the boundary of the rectangle.

Examples

			a(4) = 16 because among the five domino tilings of the 2 X 4 rectangle, 16 dominoes share a length 2 side with the boundary.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A000045.

Programs

Mathematica

Rest@ CoefficientList[Series[x (1 + 2 x - x^2 - 2 x^3 - x^4)/(1 - x - x^2)^2, {x, 0, 35}], x] (* Michael De Vlieger, Nov 05 2018 *)

PARI

Vec(x*(1 + 2*x - x^2 - 2*x^3 - x^4) / (1 - x - x^2)^2 + O(x^40)) \\ Colin Barker, Nov 02 2018

Formula

a(n) = a(n-1) + a(n-2) + 2*Fibonacci(n-1) for n > 3.

From Colin Barker, Nov 02 2018: (Start)

G.f.: x*(1 + 2*x - x^2 - 2*x^3 - x^4) / (1 - x - x^2)^2.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>5.

a(n) = 2^(1-n)*(-4*sqrt(5)*((1-sqrt(5))^n - (1+sqrt(5))^n) + 5*((1-sqrt(5))^n + (1+sqrt(5))^n)*n) / 25 for n>1.

(End)