A237356 The number of tilings of the 3 X 4 X (2n) room with 1 X 2 X 2 boxes.

1, 29, 1065, 41097, 1602289, 62603505, 2447085377, 95662064129, 3739717169185, 146197357313057, 5715321341103969, 223430193355808865, 8734601289109031137, 341463519887132765409, 13348901883923975256545, 521851299448684501083617, 20400837546324144424724449, 797534035150318477886048225, 31178158042817899845549718497

Offset: 0

Author

R. J. Mathar, Feb 07 2014

Keywords

• nonn
• easy

Comments

The count compiles all arrangements without respect to symmetry: Stacks that are equivalent after rotations or flips through any of the 3 axes or 3 planes are counted with multiplicity.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• R. J. Mathar, Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices, arXiv:1406.7788 [math.CO], eq. (44).
• Index entries for linear recurrences with constant coefficients, signature (55,-700,3186,-5180,2640).

Crossrefs

Cf. A001045 (2 X 2 X n rooms), A083066 (2 X 3 X n rooms).

Programs

Maple

A237356 := proc(n)
        (1-2*x)*(-120*x^3+122*x^2-24*x+1) / ( (1-x) *(2640*x^4-2540*x^3+646*x^2-54*x+1) ) ;
        coeftayl(%,x=0,n) ;
end proc:
seq(A237356(n),n=0..20) ;

Mathematica

CoefficientList[Series[(1 - 2 x) (-120 x^3 + 122 x^2 - 24 x + 1)/((1 - x) (2640 x^4 - 2540 x^3 + 646 x^2 - 54 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 08 2014 *)

Formula

G.f.: (1-2*x)*(-120*x^3+122*x^2-24*x+1) / ( (1-x) *(2640*x^4-2540*x^3+646*x^2-54*x+1) ) .