OFFSET
0,3
COMMENTS
a(n) <= A052980(n).
LINKS
Peter Kagey, Table of n, a(n) for n = 0..2500
Misha Lavrov, Number of ways to tile a room with I-Shaped and L-Shaped Tiles, Mathematics Stack Exchange.
Index entries for linear recurrences with constant coefficients, signature (2,-1,3,-1,2).
FORMULA
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + 2*a(n-5), with a(0) = a(1) = 1, a(2) = 2, a(3) = 5, and a(4) = 10.
G.f.: (1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5). - Colin Barker, Nov 12 2019
EXAMPLE
For n=3, the five tilings are:
+---+---+---+ +---+---+---+
| | | | | | |
+ + + + + +---+---+
| | | | | | |
+---+---+---+, +---+---+---+,
+---+---+---+ +---+---+---+
| | | | | |
+---+---+ + + +---+ +
| | | | | |
+---+---+---+, +---+---+---+, and
+---+---+---+
| | |
+ +---+ +
| | |
+---+---+---+.
For n=4, the only tiling counted by A052980(4) that is not counted by a(4) is
+---+---+---+---+
| | |
+---+---+---+---+
| | |
+---+---+---+---+.
MATHEMATICA
LinearRecurrence[{2, -1, 3, -1, 2}, {1, 1, 2, 5, 10}, 50] (* Paolo Xausa, Apr 08 2024 *)
PROG
(PARI) Vec((1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5) + O(x^30)) \\ Colin Barker, Nov 12 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Kagey, Nov 07 2019
STATUS
approved