A360644 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, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

1, 12, 513, 16194, 547543, 18234354, 609298887, 20344385080, 679408772089, 22688284005780, 757662377924917, 25301659203704234, 844933359518672599, 28216027727373068302, 942256839186226313727, 31466085716246304261600, 1050790517091131646143477

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 14.

Links

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

Index entries for linear recurrences with constant coefficients, signature (28,195,-497,30,-79,66).

Formula

G.f.: (1-16*x-18*x^2-13*x^3+10*x^4) / (1-28*x-195*x^2+497*x^3-30*x^4+79*x^5-66*x^6)

Recurrence 1:

a(n) = 12*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 43*a(n-2) + 8*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) = 60*a(n-1) + 16*b(n-1) + 6*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) = 64*a(n-1) + 13*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)=28*a(n-1) + 195*a(n-2) - 497*a(n-3) + 30*a(n-4) - 79*a(n-5) + 66*a(n-6)

for n>=6. For n<6, recurrence 1 can be used.

Crossrefs

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

Sequence in context: A197984 A197771 A198056 * A004801 A202079 A296685

Adjacent sequences: A360641 A360642 A360643 * A360645 A360646 A360647

Keyword

nonn

Author

Gerhard Kirchner, Feb 15 2023

Status

approved