A237358 The number of tilings of the 3 X 4 X n room with 1 X 2 X 3 boxes.

1, 1, 11, 64, 296, 1716, 9123, 48761, 264457, 1420548, 7652666, 41237256, 222050029, 1196138637, 6442843111, 34702528552, 186921714672, 1006820870616, 5423072856651, 29210535955209, 157337764568209, 847474515870020, 4564784961695166, 24587476389796440

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.

The rational generating function is the main body of the Maple program.

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. (58).
• Index entries for linear recurrences with constant coefficients, signature (2, 14, 42, -42, -237, -504, -103, 487, 1012, 448, -306, -74, -915, 450, -873, -54, 162).

Crossrefs

Cf. A000079 (2 X 2 X n rooms), A103143 (2 X 3 X n rooms).

Programs

Maple

A237358 := proc(n)
    (1-x)*(1+x)*(1-3*x)*(3*x^2+2*x+1)*(1-x^2-7*x^3+9*x^6)/
    (504*x^6 +306*x^11 +1 -1012*x^9 +103*x^7 -2*x +54*x^16 -162*x^17
    -450*x^14 +74*x^12 -14*x^2 -487*x^8 -42*x^3 -448*x^10 +915*x^13
    +237*x^5 +873*x^15 +42*x^4) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(A237358(n),n=0..20) ;

Mathematica

CoefficientList[Series[(1 - x) (1 + x) (1 - 3 x) (3 x^2 + 2 x + 1) (1 - x^2 - 7 x^3 + 9 x^6)/(504 x^6 + 306 x^11 + 1 - 1012 x^9 + 103 x^7 - 2 x + 54 x^16 - 162 x^17 - 450 x^14 + 74 x^12 - 14 x^2 - 487 x^8 - 42 x^3 - 448 x^10 + 915 x^13 + 237 x^5 + 873 x^15 + 42 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 08 2014 *)