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A277692 Mendelsohn-Rodney sequence: number of court balanced tournament designs that are available for a given set of teams n. 1
0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 4, 1, 5, 3, 2, 3, 6, 1, 2, 3, 6, 1, 4, 1, 4, 5, 2, 1, 8, 2, 3, 3, 4, 1, 4, 3, 6, 3, 2, 1, 8, 1, 2, 5, 6, 3, 4, 1, 4, 3, 4, 1, 9, 1, 2, 5, 4, 3, 4, 1, 8, 4, 2, 1, 8, 3, 2, 3, 6, 1, 6, 3, 4, 3, 2, 3, 10, 1, 3, 5, 6, 1, 4, 1, 6, 7, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

E. Mendelsohn and P. Rodney, The existence of court balanced tournament designs, Discrete Mathematics, 133 (1994), 207-216.

FORMULA

a(n) is the number of values of c that satisfy the following conditions: C(n,2) mod c = 0, and (n-1) mod c = 0, and 1 <= c <= floor(n/2).

a(2n) = A000005(2n-1)-1 for n > 1. - Chai Wah Wu, Oct 29 2016

EXAMPLE

For n = 9, c = 1, 2, 4 satisfy the conditions given in the formula, so a(n) = 3.

MATHEMATICA

{0}~Join~Table[Function[k, Count[Range@ k, c_ /; And[Divisible[n - 1, c], Divisible[Binomial[n, 2], c]] ]]@ Floor[n/2], {n, 2, 108}] (* Michael De Vlieger, Oct 27 2016 *)

PROG

(Python 3.5)

import scipy.stats

teams = 151

courtcount = []

i= 1

for j in range(1, teams):

    t = 0

    for i in range(1, int(j/2) + 1):

        if j>1 and ((j*(j-1))/2)%i == 0 and (j-1)%i == 0:

            t += 1

    if j > 1:

        courtcount.append(t)

a = 0

for p in courtcount:

    if p == 1:

        a+=1

print(courtcount)

(PARI) a(n) = #select(x->(!(binomial(n, 2) % x)) && !((n-1) % x), vector(n\2, k, k)); \\ Michel Marcus, Oct 27 2016

(Python)

from __future__ import division

from sympy import divisors

def A277692(n):

    return sum(1 for c in divisors(n-1) if c < n-1 and not (n*(n-1)//2) % c) if n != 2 else 1 # Chai Wah Wu, Oct 29 2016

CROSSREFS

Cf. A274332.

Sequence in context: A252665 A001055 A320266 * A129138 A112970 A112971

Adjacent sequences:  A277689 A277690 A277691 * A277693 A277694 A277695

KEYWORD

nonn

AUTHOR

Andrew G. McEachern, Oct 27 2016

STATUS

approved

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Last modified March 25 20:10 EDT 2019. Contains 321477 sequences. (Running on oeis4.)