OFFSET
1,2
COMMENTS
Number of irreducible zero-sum subsets of T(n) = {-2*n+1, -2*n+3, ..., -3, -1, 1, 3, ..., 2*n-3, 2*n-1}. A zero-sum integer set is called reducible if it contains a smaller zero-sum subset, otherwise, it is called irreducible.
The irreducible zero-sum subsets of T(n) include the following 4 categories: (a) Those inherited from T(n-1); (b) {-2*n+1, 2*n-1}; (c) those containing -2*n+1 but not 2*n-1; (d) those containing 2*n-1 but not -2*n+1. Among these, the latter 3 categories are exactly the newly added compared with T(n-1). Categories (c) and (d) are negatively symmetric, so it is sufficient to calculate the count of one of them; this count is denoted as A390083(n). Then, a(n) = a(n-1) + 1 + 2*A390083(n).
LINKS
Hu Junhua, Balanced samples of the initial segment of natural numbers, Chinese Math R & D Network BBS (in Chinese), April 21, 2010.
EXAMPLE
For n = 3, a(3) = 3, since {-5, -3, -1, 1, 3, 5} has 3 irreducible zero-sum subsets: {-1, 1}, {-3, 3}, {-5, 5}
For n = 4, a(4) = 6, since {-7, -5, -3, -1, 1, 3, 5, 7} has 6 irreducible zero-sum subsets: {-1, 1}, {-3, 3}, {-5, 5}, {-7, 7}, {-1, 3, 5, -7}, {1, -3, -5, 7}
PROG
(Python)
from itertools import combinations
def A389804(n):
if n<2: return 1
b, c = [], 0
for i in range(2, k:=2*n, 2):
j = i*(k+1)
for p in combinations(range(1, k+1), i):
if sum(p)<<1 == j:
q = set(p)
for r in b:
if r <= q:
break
else:
b.append(q)
c += 1
return c # Chai Wah Wu, Oct 23 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Hu Junhua, Oct 21 2025
EXTENSIONS
a(15)-a(18) from Sean A. Irvine, Oct 21 2025
a(19) from Chai Wah Wu, Oct 23 2025
a(20)-a(37) from Christian Sievers, Oct 25 2025
STATUS
approved