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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). 2

%I #16 Mar 03 2024 17:17:41

%S 1,11,444,13311,422617,13265660,417336617,13123557903,412719195520,

%T 12979269602143,408175860119021,12836425011761592,403683424226081169,

%U 12695147020245034099,399240466722076292612,12555423726269799691295,394846409914451855949249

%N 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).

%C 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.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (26,176,-146,-14,-140,27).

%F 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).

%F Recurrence 1:

%F 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),

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

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

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

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

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

%F Recurrence 2:

%F 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.

%o (Maxima) /* See A359884. */

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

%K nonn

%O 0,2

%A _Gerhard Kirchner_, Jan 30 2023

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Last modified March 28 16:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)