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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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Table of n, a(n) for n=0..82.
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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(C(n+1-k,k-j)*C(k-j,j),j=0..ceiling((k-1)/2)) [From Dennis Walsh, Apr 4 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); [From Dennis Walsh, Apr 4 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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Cf. A027907, A078803. See A082601 for another version.
Sequence in context: A053423 A216201 A127514 * A216232 A217765 A108482
Adjacent sequences: A078799 A078800 A078801 * A078803 A078804 A078805
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Dec 06 2002
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STATUS
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approved
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