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%I #10 Jun 26 2018 04:53:38
%S 1,1,2,2,2,3,2,4,4,3,5,5,4,6,4,7,7,5,8,5,9,9,6,10,10,7,11,7,12,12,8,
%T 13,13,9,14,9,15,15,10,16,10,17,17,11,18,18,12,19,12,20,20,13,21,13,
%U 22,22,14,23,23,15,24,15,25,25,16,26,26,17,27,17,28,28,18,29,18,30,30,19,31,31,20,32,20,33,33,21,34,34,22,35,22,36,36,23,37,23
%N Number of factorizations of the length-n prefix of the Fibonacci word A003849 into a (not strictly) decreasing sequence of finite Fibonacci words.
%C The finite Fibonacci words are the prefixes of length F(n) of A003849.
%H Pierre Bonardo, Anna E. Frid, <a href="https://arxiv.org/abs/1806.09534">Number of valid decompositions of Fibonacci prefixes</a>, arXiv:1806.09534 [math.CO], 2018.
%e For n = 7 the two factorizations of 0100101 are (01001)(01) and (010)(01)(01).
%Y Cf. A299995, which counts arbitrary factorizations, and A000119, which counts strictly decreasing factorizations.
%K nonn
%O 1,3
%A _Jeffrey Shallit_, Feb 23 2018
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