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 A078821 #13 Mar 05 2024 07:10:28 %S A078821 0,1,1,0,2,0,1,1,1,1,0,2,2,2,0,1,2,2,2,2,1,0,2,4,4,4,2,0,1,3,4,5,5,4, %T A078821 3,1,0,2,6,8,10,8,6,2,0,1,4,7,10,12,12,10,7,4,1,0,2,8,14,20,22,20,14, %U A078821 8,2,0,1,5,11,18,25,29,29,25,18,11,5,1,0,2,10,22,36,48,52,48,36,22,10,2,0 %N A078821 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's and all runlengths odd. %C A078821 Rows are symmetric. Row sums (0,2,2,4,6,10,16,26,...) are given by 2*F(n), where F(n) is the n-th Fibonacci number, A000045(n). %D A078821 Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151. %H A078821 Zachary Greenberg, Dani Kaufman, and Anna Wienhard, <a href="https://arxiv.org/abs/2402.19300">SL_2-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers</a>, arXiv:2402.19300 [math.RA], 2024. See p. 38. %F A078821 T(n, k) = A078807(n, k) + A078808(n, k). %e A078821 T(6,2) counts the words 010001, 000101, 101000 and 100010. %e A078821 Top of triangle: %e A078821 0 %e A078821 1 1 %e A078821 0 2 0 %e A078821 1 1 1 1 %e A078821 0 2 2 2 0 %e A078821 1 2 2 2 2 1 %Y A078821 Cf. A000045, A078807, A078808. %K A078821 nonn,tabl %O A078821 0,5 %A A078821 _Clark Kimberling_, Dec 07 2002