|
|
A225202
|
|
Number of aperiodic tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1.
|
|
3
|
|
|
1, 1, 2, 3, 7, 9, 20, 29, 52, 80, 143, 217, 376, 588, 977, 1563, 2583, 4116, 6764, 10854, 17688, 28512, 46367, 74763, 121385, 196040, 317756, 513616, 832039, 1345192, 2178308, 3522981, 5702741, 9224880, 14930324, 24153416, 39088168, 63239220, 102333776, 165569166, 267914295, 433476128, 701408732
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
a(n) is the Möbius transform of Fibonacci(n+1).
|
|
LINKS
|
|
|
FORMULA
|
a(p)+1 = Fibonacci(p+1) for any prime p.
|
|
EXAMPLE
|
A 4 x 1 rectangle can be tiled in 5 ways:
+-+-+-+-+ +---+-+-+ +-+---+-+ +-+-+---+ +---+---+
| | | | | | | | | | | | | | | | | | | |
+-+-+-+-+, +---+-+-+, +-+---+-+, +-+-+---+ and +---+---+.
The first tiling is 1-periodic, the last tiling is 2-periodic, while the others are not periodic. Hence a(4)=3.
Note that although the three remaining tilings are equivalent by circular shift, they are considered as distinct.
|
|
MATHEMATICA
|
a[n_] := DivisorSum[n, MoebiusMu[n/#] * Fibonacci[#+1] &]; Array[a, 50] (* Amiram Eldar, Aug 22 2023 *)
|
|
PROG
|
(PARI) a(n)=sumdiv(n, d, moebius(n/d)*fibonacci(d+1))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|