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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.

%I A320947 #30 Nov 11 2018 08:09:36
%S A320947 1,4,8,16,30,56,102,184,328,580,1018,1776,3082,5324,9160,15704,26838,
%T A320947 45736,77742,131840,223112,376844,635378,1069536,1797650,3017236,
%U A320947 5057672,8467744,14161038,23657240,39482358,65832136,109671112,182552404,303629290
%N 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.
%C A320947 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.
%H A320947 Colin Barker, <a href="/A320947/b320947.txt">Table of n, a(n) for n = 1..1000</a>
%H A320947 B. E. Tenner, <a href="https://arxiv.org/abs/1811.00082">Tiling-based models of perimeter and area</a>, arXiv:1811.00082 [math.CO], 2018.
%H A320947 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A320947 a(n) = a(n-1) + a(n-2) + 2*Fibonacci(n-1) for n > 3.
%F A320947 From _Colin Barker_, Nov 02 2018: (Start)
%F A320947 G.f.: x*(1 + 2*x - x^2 - 2*x^3 - x^4) / (1 - x - x^2)^2.
%F A320947 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>5.
%F A320947 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.
%F A320947 (End)
%e A320947 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.
%t A320947 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 *)
%o A320947 (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
%Y A320947 Cf. A000045.
%K A320947 nonn,easy
%O A320947 1,2
%A A320947 _Bridget Tenner_, Oct 24 2018