A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

1, 3, 60, 657, 8311, 101284, 1246049, 15292819, 187803572, 2305968393, 28315208039, 347681742812, 4269186204201, 52421329940803, 643681521419708, 7903765218510353, 97050331862075975, 1191681006432895780, 14632650860374551265, 179674317212728197891, 2206220907971874345652

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 13.

Links

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (8,51,27,-96,-43,66).

Formula

G.f.: (1-5*x-15*x^2-3*x^3+10*x^4) / (1-8*x-51*x^2-27*x^3+96*x^4+43*x^5-66*x^6).

Recurrence 1:

a(n) = 3*a(n-1) + b(n-1) + c(n-1) + 19*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2)

b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1)

c(n) = 20*a(n-1) + 6*b(n-1) + 2*c(n-1) + 2*e(n-1)

d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1)

e(n) = 24*a(n-1) + 7*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) = 8*a(n-1) + 51*a(n-2) + 27*a(n-3) - 96*a(n-4) - 43*a(n-5) + 66*a(n-6) for n>=6. For n<6, recurrence 1 can be used.

Mathematica

LinearRecurrence[{8, 51, 27, -96, -43, 66}, {1, 3, 60, 657, 8311, 101284}, 25] (* Paolo Xausa, Jun 01 2026 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; 66, -43, -96, 27, 51, 8]^n*[1; 3; 60; 657; 8311; 101284])[1, 1] \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360576.

Sequence in context: A283381 A002738 A006432 * A195550 A377677 A378777

Adjacent sequences: A360574 A360575 A360576 * A360578 A360579 A360580

Keyword

nonn,easy

Author

Gerhard Kirchner, Feb 12 2023

Status

approved