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%I #19 Aug 24 2023 03:13:51
%S 1,1,2,3,7,9,20,29,52,80,143,217,376,588,977,1563,2583,4116,6764,
%T 10854,17688,28512,46367,74763,121385,196040,317756,513616,832039,
%U 1345192,2178308,3522981,5702741,9224880,14930324,24153416,39088168,63239220,102333776,165569166,267914295,433476128,701408732
%N Number of aperiodic tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1.
%C a(n) is the Möbius transform of Fibonacci(n+1).
%H Paul Tek, <a href="/A225202/b225202.txt">Table of n, a(n) for n = 1..1000</a>
%H Paul Tek, <a href="/A225202/a225202_1.png">Illustration of the first terms</a>.
%F a(p)+1 = Fibonacci(p+1) for any prime p.
%e A 4 x 1 rectangle can be tiled in 5 ways:
%e +-+-+-+-+ +---+-+-+ +-+---+-+ +-+-+---+ +---+---+
%e | | | | | | | | | | | | | | | | | | | |
%e +-+-+-+-+, +---+-+-+, +-+---+-+, +-+-+---+ and +---+---+.
%e The first tiling is 1-periodic, the last tiling is 2-periodic, while the others are not periodic. Hence a(4)=3.
%e Note that although the three remaining tilings are equivalent by circular shift, they are considered as distinct.
%t a[n_] := DivisorSum[n, MoebiusMu[n/#] * Fibonacci[#+1] &]; Array[a, 50] (* _Amiram Eldar_, Aug 22 2023 *)
%o (PARI) a(n)=sumdiv(n,d,moebius(n/d)*fibonacci(d+1))
%Y Cf. A000045, A001037.
%K nonn
%O 1,3
%A _Paul Tek_, May 01 2013
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