OFFSET
0,2
COMMENTS
The first recurrence 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 5.
The example uses two cross section profiles with two overstanding cubes: C (with a common square) and D (with one common edge).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (60,-128).
FORMULA
G.f.: (1 - 16*x) / (1 - 60*x + 128*x^2).
a(n) = 44*a(n-1) + 6*e(n-1) where e(n) = 96*a(n-1) + 16*e(n-1) with a(n),e(n) <= 0 for n < =0 except for a(0)=1.
a(n) = 60*a(n-1) - 128*a(n-2) for n >= 2.
E.g.f.: exp(30*x)*cosh(2*sqrt(193)*x) + 7*exp(30*x)*sinh(2*sqrt(193)*x)/sqrt(193). - Stefano Spezia, Jan 21 2023
EXAMPLE
a(1)=44.
t1,t2,t3 is a tromino standing on 1,2,3 cubes respectively.
1) Two t2-tiles generate a C-profile or a D-profile in 4 ways each.
C,D-profile: 4,2 rotation images, D-profile: 2 ways for each image.
C-profile D-profiles
. ___ ___ ___
. /__ /| ___ /__ /| ___ /__ /|
. /__ /| |___ /__ /| | | /__ /| | |
.| | |/__ /| | | |___| | | | |___| |
.| |/__ /| | | |/__ /| | | |/__ / |
.| | |/ | | |/ | | | /
.|_______|/ |_______|/ |___|___|/
2) t1+t3 generates a C-profile in 4*2=8 ways.
. ___
. / /| ______
. /__ / | _______ /_____ /| _______
.| | / /__ /| | | | /__ /|
.| | | | /__ / | or | __|/ | /__ / |
.| | | |_| | / | | | |_| | /
.|___|/ |___|/ |___|/ |___|/
1,2) There are 12 ways to generate a C-profile. The connection of two C-profiles is a 2 X 2 X 3 cuboid. Starting with a C-profile, there are 4*3*3=36 ways to generate this cuboid.
3) There are 4*2=8 ways to generate the cuboid by starting with a D-profile. Therefore, a(1)=36+8=44.
. ___
. / /| ___ ___
. /__ / | ___ /__ /| / /|
.| | | /__ /| | | /__ / |
.|___|/| | | | |___| | | | /
. |___|/ | |/__ /| | | | | or
. | | |/ | | |
. |_______|/ |___|/
. _______
. /______ /| ___
.| | | ___ /__ /| _______
.| ___|/ /__ /| | | /______ /|
.| | | | | |___| | | | |
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. | | |/ | | |
. |_______|/ |___|/
MATHEMATICA
LinearRecurrence[{60, -128}, {1, 44}, 20] (* Paolo Xausa, Jun 24 2024 *)
PROG
(Maxima) /* See A359884. */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gerhard Kirchner, Jan 20 2023
STATUS
approved