A225202 Number of aperiodic tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1.

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

Offset: 1

Author

Paul Tek, May 01 2013

Keywords

• nonn

Comments

a(n) is the Möbius transform of Fibonacci(n+1).

Examples

			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.
		

Links

• Paul Tek, Table of n, a(n) for n = 1..1000
• Paul Tek, Illustration of the first terms.

Crossrefs

Cf. A000045, A001037.

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Fibonacci[#+1] &]; Array[a, 50] (* Amiram Eldar, Aug 22 2023 *)

PARI

a(n)=sumdiv(n,d,moebius(n/d)*fibonacci(d+1))

Formula

a(p)+1 = Fibonacci(p+1) for any prime p.