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A365303 a(n) is the smallest nonnegative integer such that the sum of any seven ordered terms a(k), k<=n (repetitions allowed), is unique.

%I A365303 #42 Mar 07 2024 17:27:43
%S A365303 0,1,8,57,256,1153,4181,14180,47381,115267,307214,737909,1682367,
%T A365303 3850940,8557010,18311575,37925058,61662056
%N A365303 a(n) is the smallest nonnegative integer such that the sum of any seven ordered terms a(k), k<=n (repetitions allowed), is unique.
%C A365303 This is the greedy B_7 sequence.
%H A365303 J. Cilleruelo and J Jimenez-Urroz, <a href="https://doi.org/10.1112/S0025579300015758">B_h[g] sequences</a>, Mathematika (47) 2000, pp. 109-115.
%H A365303 Melvyn B. Nathanson, <a href="https://arxiv.org/abs/2310.14426">The third positive element in the greedy B_h-set</a>, arXiv:2310.14426 [math.NT], 2023.
%H A365303 Melvyn B. Nathanson and Kevin O'Bryant, <a href="https://arxiv.org/abs/2311.14021">The fourth positive element in the greedy B_h-set</a>, arXiv:2311.14021 [math.NT], 2023.
%H A365303 Kevin O'Bryant, <a href="https://doi.org/10.37236/32">A complete annotated bibliography of work related to Sidon sequences</a>, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
%e A365303 a(3) != 7 because 7+0+0+0+0+0+0 = 1+1+1+1+1+1+1.
%o A365303 (Python)
%o A365303 def GreedyBh(h, seed, stopat):
%o A365303     A = [set() for _ in range(h+1)]
%o A365303     A[1] = set(seed)    # A[i] will hold the i-fold sumset
%o A365303     for j in range(2,h+1): # {2,...,h}
%o A365303         for x in A[1]:
%o A365303             A[j].update([x+y for y in A[j-1]])
%o A365303     w = max(A[1])+1
%o A365303     while w <= stopat:
%o A365303         wgood = True
%o A365303         for k in range(1,h):
%o A365303             if wgood:
%o A365303                 for j in range(k+1,h+1):
%o A365303                     if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
%o A365303                         wgood = False
%o A365303         if wgood:
%o A365303             A[1].add(w)
%o A365303             for k in range(2,h+1): # update A[k]
%o A365303                 for j in range(1,k):
%o A365303                     A[k].update([(k-j)*w + x for x in A[j]])
%o A365303         w += 1
%o A365303         return A[1]
%o A365303 GreedyBh(7,[0],10000)
%o A365303 (Python)
%o A365303 from itertools import count, islice, combinations_with_replacement
%o A365303 def A365303_gen(): # generator of terms
%o A365303     aset, alist = set(), []
%o A365303     for k in count(0):
%o A365303         bset = set()
%o A365303         for d in combinations_with_replacement(alist+[k],6):
%o A365303             if (m:=sum(d)+k) in aset:
%o A365303                 break
%o A365303             bset.add(m)
%o A365303         else:
%o A365303             yield k
%o A365303             alist.append(k)
%o A365303             aset |= bset
%o A365303 A365303_list = list(islice(A365303_gen(),10)) # _Chai Wah Wu_, Sep 01 2023
%Y A365303 Row 7 of A365515.
%Y A365303 Cf. A025582, A051912, A365300, A365301, A365302, A365304, A365305.
%K A365303 nonn,more
%O A365303 1,3
%A A365303 _Kevin O'Bryant_, Aug 31 2023
%E A365303 a(13)-a(18) from _Chai Wah Wu_, Sep 13 2023