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.

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, 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, 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

Offset: 0

Author

Clark Kimberling, Dec 07 2002

Keywords

• nonn
• tabl

Comments

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).

Examples

			T(6,2) counts the words 010001, 000101, 101000 and 100010.
Top of triangle:
  0
  1 1
  0 2 0
  1 1 1 1
  0 2 2 2 0
  1 2 2 2 2 1
		

References

• 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.

Links

• Zachary Greenberg, Dani Kaufman, and Anna Wienhard, SL_2-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers, arXiv:2402.19300 [math.RA], 2024. See p. 38.

Crossrefs

Cf. A000045, A078807, A078808.

Formula

T(n, k) = A078807(n, k) + A078808(n, k).