OFFSET
1,1
COMMENTS
{0, 1, n+1, n^2+n+1, A369817(n), a(n)} is the lexicographically first set of 6 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.
LINKS
M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
FORMULA
Conjectured that a(6n+i) is a quartic polynomial sequence with lead term (1/3)n^4 for each i in {1,2,3,5,6,10} in arxiv:2312.10910.
Proved that (1/8)*n^4 + (1/2)*n^3 <= a(n) <= 0.406671*n^4 + O(n^3) in arxiv:2312.10910.
EXAMPLE
a(2) = 20, as all 21 nonincreasing sums from {0,1,3,7,12,20}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 0+20 < 1+20 < 3+20 < 12+12 < 7+20 < 12+20 < 20+20, are distinct, and all other 6-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20}.
PROG
(Python)
from itertools import count, combinations_with_replacement
def A369818(n):
alist = [0, 1, n+1, n*(n+1)+1, (n+3>>1)*n**2+(3*n+2>>1)]
aset = set(sum(d) for d in combinations_with_replacement(alist, n))
blist = []
for i in range(n):
blist.append(set(sum(d) for d in combinations_with_replacement(alist, i)))
for k in count(max(alist[-1]+1, (n**3>>1)*(1+(n>>2)))):
for i in range(n):
if any((n-i)*k+d in aset for d in blist[i]):
break
else:
return k # Chai Wah Wu, Feb 28 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin O'Bryant, Feb 03 2024
STATUS
approved