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A008934 Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i. 25
1, 1, 2, 7, 41, 397, 6377, 171886, 7892642, 627340987, 87635138366, 21808110976027, 9780286524758582, 7981750158298108606, 11950197013167283686587, 33046443615914736611839942 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of Meeussen sequences of length n (see the Cook-Kleber reference).

Column 1 of triangle A093729. Also generated by the iteration procedure that constructs triangle A093654. - Paul D. Hanna, Apr 14 2004

LINKS

T. D. Noe, Table of n, a(n) for n=0..30

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

E. Neuwirth, Computing tournament sequence numbers efficiently..., Séminaire Lotharingien de Combinatoire, B47h (2002), 12 pp.

Mauro Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 40:2 (2006), pp. 107-121.

Index entries for sequences related to tournaments

FORMULA

a(n) = A093729(n, 1). a(n) = A093655(2^n). - Paul D. Hanna, Apr 14 2004

a(n) = A097710(n, 0). - Paul D. Hanna, Aug 24 2004

From Benedict W. J. Irwin, Nov 26 2016: (Start)

Conjecture: a(n) is given by a series of nested sums as follows:

a(2) = Sum_{i=1..2} 1,

a(3) = Sum_{i=1..2}Sum_{j=1..i+2} 1,

a(4) = Sum_{i=1..2}Sum_{j=1..i+2}Sum_{k=1..i+j+2} 1,

a(5) = Sum_{i=1..2}Sum_{j=1..i+2}Sum_{k=1..i+j+2}Sum_{l=1..i+j+k+2} 1.

(End)

MATHEMATICA

t[n_?Negative, _] = 0; t[0, _] = 1; t[_, 0] = 0; t[n_, k_] /; k <= n :=  t[n, k] = t[n, k-1] - t[n-1, k] + t[n-1, 2k-1] + t[n-1, 2 k]; t[n_, k_] /; k > n :=  t[n, k] =Sum[(-1)^(j-1) Binomial[n+1, j]*t[n, k-j] , {j, 1, n+1}]; Table[t[n, 1], {n, 0, 15} ] (* Jean-François Alcover, May 17 2011, after PARI prog. *)

PROG

(PARI) {T(n, k)=if(n<0, 0, if(n==0, 1, if(k==0, 0, if(k<=n, T(n, k-1)-T(n-1, k)+T(n-1, 2*k-1)+T(n-1, 2*k), sum(j=1, n+1, (-1)^(j-1)*binomial(n+1, j)*T(n, k-j))))))} /*(Cook-Kleber)*/ a(n)=T(n, 1)

CROSSREFS

Cf. A058222, A058223.

Cf. A093729, A093655.

Forms column 0 of triangle A097710.

Sequence in context: A173916 A163921 A213434 * A084871 A122942 A159315

Adjacent sequences:  A008931 A008932 A008933 * A008935 A008936 A008937

KEYWORD

nonn,nice,easy

AUTHOR

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it), Jeffrey Shallit

STATUS

approved

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Last modified October 14 01:36 EDT 2019. Contains 327994 sequences. (Running on oeis4.)