OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Journal of Integer Sequences 24 (2021), article 21.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,3,5,-4).
FORMULA
a(n) = a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4).
G.f.: (1 - x)/(1 - x - 3*x^2 - 5*x^3 + 4*x^4).
EXAMPLE
Here are two such tilings for a 3 X 3 strip; each has four rotations thus demonstrating that a(3)=8.
._____. ._____.
| | | | | |___|
| |_|_| | |___|
|_____| |_____|
For a 3 X 4 strip, here are three of the possible a(4)=13 tilings.
._______. ._______. ._______.
| |___ | | ___| | |___|___|
| |___| | | |___| | | |___| |
|_____|_| |_|_____| |_|___|_|
For a 3 X 5 strip, here are three of the possible a(5)=52 tilings.
._________. ._________. ._________.
| | |___| | | ___|___| | |___|___|
| |_|___|_| | | |___| | | |___|___|
|_____|___| |_|_|___|_| |_____|___|
MATHEMATICA
LinearRecurrence[{1, 3, 5, -4}, {1, 0, 3, 8}, 50];
PROG
(Magma) I:=[1, 0, 3, 8]; [n le 4 select I[n] else Self(n-1) +3*Self(n-2) +5*Self(n-3) -4*Self(n-4): n in [1..50]]; // G. C. Greubel, Dec 01 2022
(SageMath)
@CachedFunction
def a(n): # a = A346054
if (n<4): return (1, 0, 3, 8)[n]
else: return a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Dec 01 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Greg Dresden and Ziyao Geng, Jul 02 2021
EXTENSIONS
Corrected by Greg Dresden, Sep 04 2021
STATUS
approved