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A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)*(x^3+2*x^2+x-1) ). 2

%I

%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.

%H G. C. Greubel, <a href="/A233247/b233247.txt">Table of n, a(n) for n = 0..1000</a>

%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.

%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) 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

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Last modified March 4 08:38 EST 2021. Contains 341781 sequences. (Running on oeis4.)