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

1, 1, 1, 4, 9, 16, 36, 81, 169, 361, 784, 1681, 3600, 7744, 16641, 35721, 76729, 164836, 354025, 760384, 1633284, 3508129, 7535025, 16184529, 34762816, 74666881, 160376896, 344473600, 739894401, 1589218225, 3413480625, 7331811876, 15747991081, 33825095056

Offset: 0

Author

R. J. Mathar, Dec 06 2013

Keywords

• easy
• nonn

Comments

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).

The inverse INVERT transform is 1, 0, 3, 2, 2, 4, 4, 6, 8, 10, .. , continued as in A068924.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
• R. J. Mathar, Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, see eq. (39).
• Index entries for linear recurrences with constant coefficients, signature (1,1,3,1,-1,-1).

Crossrefs

Cf. A000930.

Programs

Maple

A233247 := proc(n)
    A000930(n)^2 ;
end proc:
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[3, 3]^2:
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2013

Mathematica

Table[Sum[Binomial[n-2i, i], {i,0,n/3}]^2, {n,0,50}] (* Wesley Ivan Hurt, Dec 06 2013 *)
LinearRecurrence[{1,1,3,1,-1,-1},{1,1,1,4,9,16},40] (* Harvey P. Dale, Jan 14 2015 *)
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 *)

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

Formula

a(n) = A000930(n)^2.

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