OFFSET
1,3
COMMENTS
It is easy to see that {1,2,...,2n-1} can be replaced by any 2n-1 consecutive numbers and the results will be the same. Erdos, Ginzburg and Ziv proved that every set of 2n-1 numbers -- not necessarily consecutive -- contains a subset of n elements whose sum is a multiple of n.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1669 (terms 1..200 from T. D. Noe)
Max Alekseyev, Proof of Jovovic's formula, 2008.
Michal Bassan, Serte Donderwinkel, and Brett Kolesnik, Tournament score sequences, Erdős-Ginzburg-Ziv numbers, and the Lévy-Khintchine method, arXiv:2407.01441 [math.CO], 2024. See p. 7.
Shane Chern, An extension of a formula of Jovovic, Integers (2019) Vol. 19, Article A47.
Anders Claesson, Mark Dukes, Atli Fannar Franklín, and Sigurður Örn Stefánsson, Counting tournament score sequences, arXiv:2209.03925 [math.CO], 2022.
P. Erdős, A. Ginzburg and A. Ziv, Theorem in the additive number theory, Bull. Res. Council Israel 10 (1961).
Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, arXiv:1809.05732 [math.AG], 2018.
Mithat Ünsal, Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD, arXiv:2104.12352 [hep-th], 2021.
FORMULA
a(n) = (1/(2*n))*Sum_{d|n} (-1)^(n+d)*phi(n/d)*binomial(2*d,d). Conjectured by Vladeta Jovovic, Oct 22 2008; proved by Max Alekseyev, Oct 23 2008 (see link).
a(2n+1) = A003239(2n+1) and a(2n) = A003239(2n) - A003239(d), where d is the largest odd divisor of n. - T. D. Noe, Oct 24 2008
a(n) = Sum_{d|n} (-1)^(n+d)*d*A131868(d). - Vladeta Jovovic, Oct 28 2008
a(n) = Sum_{k=0..[n/2]} A227532(n,n*k), where A227532 is the logarithmic derivative, wrt x, of the g.f. G(x,q) = 1 + x*G(q*x,q)*G(x,q) of triangle A227543. - Paul D. Hanna, Jul 17 2013
Logarithmic derivative of A000571, the number of different scores that are possible in an n-team round-robin tournament. - Paul D. Hanna, Jul 17 2013
G.f.: -Sum_{m >= 1} (phi(m)/m) * log((1 + sqrt(1 + 4*(-y)^m))/2). - Petros Hadjicostas, Jul 15 2019
a(n) ~ 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 28 2023
EXAMPLE
a(3)=4 because, of the 10 3-element subsets of 1..7, only {1,2,3}, {1,3,5}, {2,3,4} and {3,4,5} have sums that are multiples of 3.
L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 9*x^4/4 + 26*x^5/5 + 76*x^6/6 + 246*x^7/7 +...
where exponentiation yields the g.f. of A000571:
exp(L(x)) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 22*x^6 + 59*x^7 + 167*x^8 +...
MATHEMATICA
Table[Length[Select[Plus@@@Subsets[Range[2n-1], {n}], Mod[ #, n]==0&]], {n, 10}]
Table[d=Divisors[n]; Sum[(-1)^(n+d[[i]]) EulerPhi[n/d[[i]]] Binomial[2d[[i]], d[[i]]]/2/n, {i, Length[d]}], {n, 30}] (* T. D. Noe, Oct 24 2008 *)
PROG
(PARI) {a(n)=sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/(2*n))}
(PARI) {A227532(n, k)=local(G=1); for(i=1, n, G=1+x*subst(G, x, q*x)*G +x*O(x^n)); n*polcoeff(polcoeff(log(G), n, x), k, q)}
{a(n)=sum(k=0, n\2, A227532(n, n*k))} \\ Paul D. Hanna, Jul 17 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Oct 21 2008, Oct 22 2008, Oct 24 2008
EXTENSIONS
Extension T. D. Noe, Oct 24 2008
STATUS
approved