OFFSET
0,4
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).
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
R. J. Mathar, Dec 06 2013
STATUS
approved