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%I #29 Mar 19 2024 10:25:30
%S 4,6,16,34,84,198,480,1154,2788,6726,16240,39202,94644,228486,551616,
%T 1331714,3215044,7761798,18738640,45239074,109216788,263672646,
%U 636562080,1536796802,3710155684
%N a(1)=4, a(2)=6; for n > 2, a(n) = 2*a(n-1) + a(n-2) - 4*((n-1) mod 2).
%C a(n) is the number of perfect matchings of an edge-labeled 2 X n Klein bottle grid graph, or equivalently the number of domino tilings of a 2 X n Klein bottle grid. (The twist is on the length-2 side.)
%H Sarah-Marie Belcastro, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Belcastro/belcastro4.html">Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces</a>, J. Integer Seq. 26 (2023), Article 23.5.6.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F For n > 1, a(n) = (1/2)*((1 + sqrt(2))^n*(2 + (-1 + sqrt(2))^(2*floor((1/2)*(-1 + n)))*(-4 + 3*sqrt(2))) + (1 - sqrt(2))^n*(2 - (-1 - sqrt(2))^(2*floor((1/2)*(-1 + n)))*(4 + 3*sqrt(2)))).
%F From _Colin Barker_, May 01 2012: (Start)
%F a(n) = 1 - (-1)^n + (1-sqrt(2))^n + (1+sqrt(2))^n.
%F G.f.: 2*x*(2-x-2*x^2-x^3)/(1-x)/(1+x)/(1-2*x-x^2). (End)
%F a(n) = A002203(n) + 1 - (-1)^n. - _R. J. Mathar_, Oct 08 2016
%e a(3) = 2*a(2) + a(1) - 4*(2 mod 2) = 2*6 + 4 - 0 = 16.
%K easy,nonn
%O 1,1
%A _Sarah-Marie Belcastro_, Jul 04 2009
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