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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.
4
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.
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
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