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A078802
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Triangular array T given by T(n,k) = number of 01-words of length n containing k 1's, no three of which are consecutive.
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3
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1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 2, 0, 1, 5, 10, 7, 1, 0, 1, 6, 15, 16, 6, 0, 0, 1, 7, 21, 30, 19, 3, 0, 0, 1, 8, 28, 50, 45, 16, 1, 0, 0, 1, 9, 36, 77, 90, 51, 10, 0, 0, 0, 1, 10, 45, 112, 161, 126, 45, 4, 0, 0, 0, 1, 11, 55, 156, 266, 266, 141, 30, 1, 0, 0, 0, 1, 12, 66, 210, 414
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OFFSET
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0,5
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COMMENTS
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The rows of T are essentially the antidiagonals of A027907 (trinomial coefficients). Reversing the rows produces A078803. Row sums: A000073.
Also, the diagonals of T are essentially the rows of A027907, so diagonal sums = 3^n. Antidiagonal sums are essentially A060961 (number of ordered partitions of n into 1's, 3's and 5's). - Gerald McGarvey, May 13 2005
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REFERENCES
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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.
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LINKS
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FORMULA
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T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-2) with initial values as in first 3 rows.
T(n,k) = Sum_{j=0..ceiling((k-1)/2)} C(n+1-k, k-j)*C(k-j, j). - Dennis P. Walsh, Apr 04 2012
G.f.: (1 + y*x + y^2*x^2)/(1 - (x*(1 + y*x + y^2*x^2))). - Geoffrey Critzer, Sep 15 2012
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EXAMPLE
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T(4,3) = 2 counts 1+0+1+1 and 1+1+0+1. Top of triangle T:
1;
1, 1;
1, 2, 1;
1, 3, 3, 0;
1, 4, 6, 2, 0;
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MAPLE
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seq(seq(sum(binomial(n+1-k, k-j)*binomial(k-j, j), j=0..ceil((k-1)/2)), k=0..n), n=0..20); # Dennis P. Walsh, Apr 04 2012
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MATHEMATICA
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nn=15; a=1+y x+y^2 x^2; f[list_]:=Select[list, #>0&]; Map[f, CoefficientList[Series[a/(1-x a), {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Sep 15 2012 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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