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A224918 Number of tilings of an n X 1 rectangle (using tiles of dimension 1 X 1 and 2 X 1) that are not the concatenation of smaller equally-sized tilings.

%I A224918 #27 May 06 2013 13:22:22
%S A224918 1,1,2,1,7,0,20,9,28,9,143,39,376,105,340,441,2583,480,6764,2400,7235,
%T A224918 6897,46367,10332,88625,50193,151436,126504,832039,127431,2178308,
%U A224918 974169,2618488,2484873,9209899,3202560,39088168,17218617,47865787,33738201,267914295,49047180,701408732,303913896,624579100
%N A224918 Number of tilings of an n X 1 rectangle (using tiles of dimension 1 X 1 and 2 X 1) that are not the concatenation of smaller equally-sized tilings.
%C A224918 a(p)+1 = Fibonacci(p+1) for any prime p.
%C A224918 a(2^k) = Fibonacci(2^(k-1))^2 for k>0.
%C A224918 a(n) <= A225202(n).
%H A224918 Paul Tek, <a href="/A224918/b224918.txt">Table of n, a(n) for n = 1..1000</a>
%H A224918 Paul Tek, <a href="/A224918/a224918.png">Illustration of the first terms</a>
%H A224918 Paul Tek, <a href="/A224918/a224918.txt">PERL program for this sequence</a>
%e A224918 A 4 x 1 rectangle can be tiled in 5 ways:
%e A224918   +-+-+-+-+                               +-+  +-+  +-+     +-+
%e A224918 - | | | | | that is the concatenation of  | |, | |, | | and | |
%e A224918   +-+-+-+-+                               +-+  +-+  +-+     +-+,
%e A224918   +---+-+-+                               +---+     +-+-+
%e A224918 - |   | | | that is the concatenation of  |   | and | | |
%e A224918   +---+-+-+                               +---+     +-+-+,
%e A224918   +-+---+-+
%e A224918 - | |   | | that is not the concatenation of smaller equally sized tilings,
%e A224918   +-+---+-+
%e A224918   +-+-+---+                               +-+-+     +---+
%e A224918 - | | |   | that is the concatenation of  | | | and |   |
%e A224918   +-+-+---+                               +-+-+     +---+,
%e A224918   +---+---+                               +---+     +---+
%e A224918 - |   |   | that is the concatenation of  |   | and |   |
%e A224918   +---+---+                               +---+     +---+.
%e A224918 Hence a(4)=1.
%Y A224918 Cf. A000045 (Fibonacci numbers).
%Y A224918 Cf. A225202.
%K A224918 nonn
%O A224918 1,3
%A A224918 _Paul Tek_, May 04 2013