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 A129717 #15 Dec 29 2023 10:56:07
%S A129717 1,2,3,4,1,4,4,4,8,1,4,12,5,4,16,13,1,4,20,25,6,4,24,41,19,1,4,28,61,
%T A129717 44,7,4,32,85,85,26,1,4,36,113,146,70,8,4,40,145,231,155,34,1,4,44,
%U A129717 181,344,301,104,9,4,48,221,489,532,259,43,1,4,52,265,670,876,560,147,10,4,56
%N A129717 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 101's (n >= 0, 0 <= k <= floor((n-1)/2)). A Fibonacci binary word is a binary word having no 00 subword.
%C A129717 Row n has 1+floor((n-1)/2) terms for n >= 1.
%C A129717 Row sums are the Fibonacci numbers (A000045).
%C A129717 T(n,1) = A008574(n-3).
%C A129717 T(n,2) = A001844(n-5).
%C A129717 T(n,3) = A005900(n-6).
%C A129717 T(n,4) = A006325(n-7).
%C A129717 T(n,5) = A033455(n-10).
%C A129717 T(n,k) = A129718(n,k+1) (since in each word: 1 + the number of 101's = number of runs of 1's).
%C A129717 Sum_{k>=0} k*T(n,k) = A004798(n-2).
%H A129717 Michael De Vlieger, <a href="/A129717/b129717.txt">Table of n, a(n) for n = 0..10100</a> (rows 0 <= n <= 200, flattened.)
%F A129717 G.f.: G(t,z) = (1+z)*(1 + z^2 - t*z^2)/(1 - z - t*z^2).
%F A129717 G.f. of col 0: (1+z)(1+z^2)/(1-z), leading to the partial sums of 1,1,1,1,0,0,0,...
%F A129717 G.f. of col k: z^(2k+1)*(1+z)^2/(1-z)^(k+1) (k >= 1).
%F A129717 T(n,k) = binomial(n-k-1, k) + 2*binomial(n-k-2, k) + binomial(n-k-3, k) for n >= 4 and 0 <= k < n/2.
%e A129717 T(6,2)=5 because we have 110101, 101101, 101010, 101011 and 010101.
%e A129717 Triangle starts:
%e A129717 1;
%e A129717 2;
%e A129717 3;
%e A129717 4, 1;
%e A129717 4, 4;
%e A129717 4, 8, 1;
%e A129717 4, 12, 5;
%p A129717 T:=proc(n,k) if n=0 and k=0 then 1 elif n=1 and k=0 then 2 elif n=2 and k=0 then 3 elif n=3 and k=1 then 1 elif k<n/2 then binomial(n-k-1,k)+2*binomial(n-k-2,k)+binomial(n-k-3,k) else 0 fi end: 1; for n from 1 to 18 do seq(T(n,k),k=0..floor(n-1)/2) od; # yields sequence in triangular form
%t A129717 MapAt[{0, 1} + # &, #, 4] /. {} -> {1} &@ Table[If[n < 3, n + 1, Binomial[n - k - 1, k] + 2 Binomial[n - k - 2, k] + Binomial[n - k - 3, k]], {n, 0, 17}, {k, 0, Floor[(n - 1)/2]}] // Flatten (* _Michael De Vlieger_, Nov 15 2019 *)
%Y A129717 Cf. A000045, A008574, A001844, A005900, A006325, A033455, A129718, A004798.
%K A129717 nonn,tabf
%O A129717 0,2
%A A129717 _Emeric Deutsch_, May 12 2007