A360644 - OEIS

%I #5 Feb 16 2023 05:14:31

%S 1,12,513,16194,547543,18234354,609298887,20344385080,679408772089,

%T 22688284005780,757662377924917,25301659203704234,844933359518672599,

%U 28216027727373068302,942256839186226313727,31466085716246304261600,1050790517091131646143477

%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, 2 X 2 X 1 plates 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 14.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (28,195,-497,30,-79,66).

%F 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)

%F Recurrence 1:

%F 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)

%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) = 60*a(n-1) + 16*b(n-1) + 6*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) = 64*a(n-1) + 13*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)=28*a(n-1) + 195*a(n-2) - 497*a(n-3) + 30*a(n-4) - 79*a(n-5) + 66*a(n-6)

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

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

%K nonn

%O 0,2

%A _Gerhard Kirchner_, Feb 15 2023