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%I #28 Sep 30 2024 15:13:09
%S 1,1,1,4,9,16,36,81,169,361,784,1681,3600,7744,16641,35721,76729,
%T 164836,354025,760384,1633284,3508129,7535025,16184529,34762816,
%U 74666881,160376896,344473600,739894401,1589218225,3413480625,7331811876,15747991081,33825095056
%N Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)*(x^3+2*x^2+x-1) ).
%C a(n) is the number of tilings of a 3 X 2 X n room with bricks of 1 X 1 X 3 shape (and in that respect a generalization of A028447 which fills 3 X 2 X n rooms with bricks of 1 X 1 X 2 shape).
%C The inverse INVERT transform is 1, 0, 3, 2, 2, 4, 4, 6, 8, 10, .. , continued as in A068924.
%C a(n) is the number of tilings of an n-board (a board with dimensions n X 1) with half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2;3)-combs. A (w,g;m)-comb is a tile composed of m pieces of dimensions w X 1 separated horizontally by gaps of width g. - _Michael A. Allen_, Sep 24 2024
%H G. C. Greubel, <a href="/A233247/b233247.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://doi.org/10.1080/03081087.2022.2107979">Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices</a>, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices</a>, arXiv:1406.7788 [math.CO], 2014, see eq. (39).
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,3,1,-1,-1).
%F a(n) = A000930(n)^2.
%F a(n) = a(n-1) + a(n-3) + 2*Sum_{r=3..n} ( A000931(r+2)*a(n-r) ). - _Michael A. Allen_, Sep 24 2024
%p A233247 := proc(n)
%p A000930(n)^2 ;
%p end proc:
%p # second Maple program:
%p a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[3, 3]^2:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Dec 06 2013
%t Table[Sum[Binomial[n-2i, i], {i,0,n/3}]^2, {n,0,50}] (* _Wesley Ivan Hurt_, Dec 06 2013 *)
%t LinearRecurrence[{1,1,3,1,-1,-1},{1,1,1,4,9,16},40] (* _Harvey P. Dale_, Jan 14 2015 *)
%t CoefficientList[Series[(1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1)), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 29 2017 *)
%o (PARI) my(x='x+O('x^50)); Vec((1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1))) \\ _G. C. Greubel_, Apr 29 2017
%Y Cf. A000930.
%K easy,nonn
%O 0,4
%A _R. J. Mathar_, Dec 06 2013