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%I #12 Jun 23 2018 05:45:33
%S 0,1,1,4,5,14,19,46,65,145,210,444,654,1331,1985,3926,5911,11434,
%T 17345,32960,50305,94211,144516,267384,411900,754309,1166209,2116936,
%U 3283145,5914310,9197455,16458034,25655489,45638101,71293590,126159156
%N Number of 0's in odd position in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.
%H É. Czabarka, R. Flórez, L. Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.
%F G.f.: z(1-z^2)/((1-z-z^2)^2*(1+z-z^2)).
%F a(2n) = a(2n-1) + a(2n-2) (n >= 1).
%F a(2n-1) = A030267(n).
%F a(2n) = A129722(2n) = A001870(n-1).
%F a(n) = Sum_{k=0..ceiling(n/2)} k*A129719(n,k).
%e a(4)=5 because in 1110, 1111, 110'1, 1010, 1011, 0'110, 0'111 and 0'10'1 one has altogether five 0's in odd position (marked by ').
%p g:=z*(1-z^2)/(1-z-z^2)^2/(1+z-z^2): gser:=series(g,z=0,43): seq(coeff(gser,z,n),n=0..40);
%Y Cf. A030267, A001870, A129719, A129722.
%K nonn
%O 0,4
%A _Emeric Deutsch_, May 13 2007