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

LINKS

Paul Tek, Table of n, a(n) for n = 1..1000

Paul Tek, Illustration of the first terms

FORMULA

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

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.

PROG

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

CROSSREFS

Cf. A000045, A001037.

Sequence in context: A204520 A007649 A218909 * A046152 A057239 A319911

Adjacent sequences:  A225199 A225200 A225201 * A225203 A225204 A225205

KEYWORD

nonn

AUTHOR

Paul Tek, May 01 2013

STATUS

approved

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Last modified June 25 12:01 EDT 2019. Contains 324352 sequences. (Running on oeis4.)