A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

1, 6, 122, 1768, 28844, 457592, 7318760, 116806896, 1865305376, 29782666544, 475549098160, 7593154541264, 121241257906000, 1935879286697296, 30910512661708432, 493553365105565264, 7880649886335326608, 125831666350680625104, 2009175449141135367696, 32080843409673995438928

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 12.

Links

Paolo Xausa, Table of n, a(n) for n = 0..800

Index entries for linear recurrences with constant coefficients, signature (15,28,-214,192,384,-128).

Formula

G.f.: (1-9*x+4*x^2-16*x^3) / (1-15*x-28*x^2+214*x^3-192*x^4-384*x^5+128*x^6).

Recurrence 1:

a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)

b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)

c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)

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

e(n) = 12*a(n-1) + 3*b(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) = 15*a(n-1) + 28*a(n-2) - 214*a(n-3) + 192*a(n-4) + 384*a(n-5) - 128*a(n-6) for n>=6. For n<6, recurrence 1 can be used.

Mathematica

LinearRecurrence[{15, 28, -214, 192, 384, -128}, {1, 6, 122, 1768, 28844, 457592}, 20] (* 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; -128, 384, 192, -214, 28, 15]^n*[1; 6; 122; 1768; 28844; 457592])[1, 1] \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360577.

Sequence in context: A393693 A296801 A249168 * A138572 A393626 A356199

Adjacent sequences: A360573 A360574 A360575 * A360577 A360578 A360579

Keyword

nonn,easy

Author

Gerhard Kirchner, Feb 12 2023

Extensions

More terms from Paolo Xausa, Jun 01 2026

Status

approved