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 A078802 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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 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 FORMULA 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 EXAMPLE 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 MAPLE 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] MATHEMATICA 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 *) CROSSREFS 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 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Dec 06 2002 STATUS approved

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Last modified May 19 00:49 EDT 2013. Contains 225428 sequences.