A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 0, 0, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 0, 0, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 0, 0, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 0, 0

Offset

1,12

Comments

Row sums: 1,1,2,3,5,8,13,..., the Fibonacci numbers (A000045).

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

Table of n, a(n) for n=1..92.

Formula

T(n, k)=T(n-1, n-k-1)+T(n-3, n-k-3)+...+T(n-2m-1, n-k-2m-1), where m=[(n-1)/2] and (by definition) T(i, j)=0 if i<0 or j<0 or i=j.

Example

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

Crossrefs

Cf. A078808, A078821, A079487, A123245, A077419.

Sequence in context: A137608 A191336 A277349 * A208249 A329985 A029422

Adjacent sequences: A078804 A078805 A078806 * A078808 A078809 A078810

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 07 2002

Extensions

Row 0 removed to stick to the triangle format by Andrey Zabolotskiy, Sep 22 2017

Status

approved