login
A362298
Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.
2
1, 1, 19, 55, 472, 2023, 13249, 66325, 392299, 2088856, 11877025, 64803157, 362823607, 1998759703, 11123273896, 61509329983, 341492705365, 1891193243713, 10489893539203, 58127214942544, 322296397820593, 1786338231961609, 9903234373856059, 54893955008138983
OFFSET
0,3
COMMENTS
Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.
FORMULA
a(n) = 4*a(n-1) + 18*a(n-2) - 48*a(n-3) - 42*a(n-4) + 99*a(n-5).
G.f.: (9*x^3-3*x^2-3*x+1)/(-99*x^5+42*x^4+48*x^3-18*x^2-4*x+1).
EXAMPLE
a(2) = 19.
Partitions of a 2 X 2 square (triangles or dominos):
___ ___ ___ ___
| /| |\ | |___| | | |
|/__| |__\| |___| |_|_|
2t 2d
___ ___ ___ ___ ___ ___ _ ___ _ _______
|2t |2t | |2t |2d | |2d |2t | | |2t | | |only d |
|___|___| |___|___| |___|___| |_|___|_| |_______|
4 ways + 4 ways + 4 ways + 2 ways + 5 ways = 19 ways
Only dominos: A005178(3) = 5.
MATHEMATICA
LinearRecurrence[{4, 18, -48, -42, 99}, {1, 1, 19, 55, 472}, 24] (* Stefano Spezia, Apr 20 2023 *)
CROSSREFS
Column k=2 of A362297.
Sequence in context: A124712 A126373 A125818 * A367518 A093362 A341176
KEYWORD
nonn,easy
AUTHOR
Gerhard Kirchner, Apr 19 2023
STATUS
approved