This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129722 #34 Jan 02 2021 04:22:01
%S A129722 0,0,1,1,5,6,19,25,65,90,210,300,654,954,1985,2939,5911,8850,17345,
%T A129722 26195,50305,76500,144516,221016,411900,632916,1166209,1799125,
%U A129722 3283145,5082270,9197455,14279725,25655489,39935214,71293590,111228804,197452746,308681550
%N A129722 Number of 0's in even position in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.
%H A129722 G. C. Greubel, <a href="/A129722/b129722.txt">Table of n, a(n) for n = 0..1000</a>
%H A129722 Moussa Benoumhani, <a href="http://www.emis.de/journals/JIS/VOL15/Benoumhani/benoumhani8.html">On the Modes of the Independence Polynomial of the Centipede</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1.
%H A129722 É. Czabarka, R. Flórez, and 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.
%H A129722 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-4,1,1).
%F A129722 G.f.: z^2/( (1+z-z^2)*(1-z-z^2)^2 ).
%F A129722 a(2*n+1) = a(2*n) + a(2*n-1) (n>=1).
%F A129722 a(2*n+1) = A001871(n-1) (n>=1).
%F A129722 a(2*n) = A129720(2*n) = A001870(n-1).
%F A129722 a(n) = Sum_{ k=0..floor(n/2)} k*A129721(n,k).
%F A129722 a(n) = F(n)*(n+1)/5 + F(n+1)*(2*n - 5 + 5*(-1)^n)/20 where F = A000045. - _Greg Dresden_, Jan 01 2021
%e A129722 a(4)=5 because in 1110', 1111, 1101, 10'10', 10'11, 0110', 0111 and 0101 one has altogether five 0's in even position (marked by ').
%p A129722 G:=z^2/(1-z-z^2)^2/(1+z-z^2): Gser:=series(G,z=0,45): seq(coeff(Gser,z,n),n=0..42);
%t A129722 CoefficientList[Series[x^2/((1 + x - x^2)*(1 - x - x^2)^2), {x,0,50}], x] (* _G. C. Greubel_, Mar 09 2017 *)
%t A129722 LinearRecurrence[{1,4,-3,-4,1,1},{0,0,1,1,5,6},40] (* _Harvey P. Dale_, Apr 02 2018 *)
%o A129722 (PARI) x='x+O('x^50); concat([0,0], Vec(x^2/((1 + x - x^2)*(1 - x - x^2)^2))) \\ _G. C. Greubel_, Mar 09 2017
%Y A129722 Cf. A000045, A001870, A001871, A129719, A129720, A129721.
%K A129722 nonn
%O A129722 0,5
%A A129722 _Emeric Deutsch_, May 13 2007