A360066 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

1, 11, 444, 13311, 422617, 13265660, 417336617, 13123557903, 412719195520, 12979269602143, 408175860119021, 12836425011761592, 403683424226081169, 12695147020245034099, 399240466722076292612, 12555423726269799691295, 394846409914451855949249

Offset

0,2

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 10.

Links

Table of n, a(n) for n=0..16.

Index entries for linear recurrences with constant coefficients, signature (26,176,-146,-14,-140,27).

Formula

G.f.: (1 - 15*x - 18*x^2 - 23*x^3 + 7*x^4) / (1 - 26*x - 176*x^2 + 146*x^3 + 14*x^4 + 140*x^5 - 27*x^6).

Recurrence 1:

a(n) = 11*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 29*a(n-2) + 6*b(n-2) + c(n-2) + 2*d(n-2),

b(n) = 32*a(n-1) + 9*b(n-1) + 4*c(n-1) + 2*d(n-1) + e(n-1),

c(n) = 52*a(n-1) + 14*b(n-1) + 5*c(n-1) + 4*d(n-1) + 2*e(n-1),

d(n) = 14*a(n-1) + 3*b(n-1) + d(n-1),

e(n) = 48*a(n-1) + 11*b(n-1) + 2*c(n-1) + 2*d(n-1),

with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.

Recurrence 2:

a(n) = 26*a(n-1) + 176*a(n-2) - 146*a(n-3) - 14*a(n-4) - 140*a(n-5) + 27*a(n-6) for n >= 6. For n < 6, recurrence 1 can be used.

Prog

(Maxima) /* See A359884. */

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065.

Sequence in context: A356210 A140840 A175158 * A354439 A180087 A233219

Adjacent sequences: A360063 A360064 A360065 * A360067 A360068 A360069

Keyword

nonn

Author

Gerhard Kirchner, Jan 30 2023

Status

approved