A360577 - OEIS

%I #14 Jun 01 2026 16:55:10

%S 1,3,60,657,8311,101284,1246049,15292819,187803572,2305968393,

%T 28315208039,347681742812,4269186204201,52421329940803,

%U 643681521419708,7903765218510353,97050331862075975,1191681006432895780,14632650860374551265,179674317212728197891,2206220907971874345652

%N 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).

%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 13.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,51,27,-96,-43,66).

%F 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).

%F Recurrence 1:

%F 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)

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

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

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

%F e(n) = 24*a(n-1) + 7*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) = 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.

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

%o (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

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

%K nonn,easy

%O 0,2

%A _Gerhard Kirchner_, Feb 12 2023