OFFSET
1,3
COMMENTS
This is the greedy B_4 sequence.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..50
J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn 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, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
EXAMPLE
a(4) != 12 because 12+1+1+1 = 5+5+5+0.
PROG
(Python)
def GreedyBh(h, seed, stopat):
A = [set() for _ in range(h+1)]
A[1] = set(seed) # A[i] will hold the i-fold sumset
for j in range(2, h+1): # {2, ..., h}
for x in A[1]:
A[j].update([x+y for y in A[j-1]])
w = max(A[1])+1
while w <= stopat:
wgood = True
for k in range(1, h):
if wgood:
for j in range(k+1, h+1):
if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
wgood = False
if wgood:
A[1].add(w)
for k in range(2, h+1): # update A[k]
for j in range(1, k):
A[k].update([(k-j)*w + x for x in A[j]])
w += 1
return A[1]
GreedyBh(4, [0], 10000)
(Python)
from itertools import count, islice, combinations_with_replacement
def A365300_gen(): # generator of terms
aset, alist = set(), []
for k in count(0):
bset = set()
for d in combinations_with_replacement(alist+[k], 3):
if (m:=sum(d)+k) in aset:
break
bset.add(m)
else:
yield k
alist.append(k)
aset |= bset
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin O'Bryant, Aug 31 2023
EXTENSIONS
a(27)-a(37) from Chai Wah Wu, Sep 01 2023
STATUS
approved