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A365300 a(n) is the smallest nonnegative integer such that the sum of any four ordered terms a(k), k<=n (repetitions allowed), is unique.

%I A365300 #49 Mar 28 2024 04:13:20
%S A365300 0,1,5,21,55,153,368,856,1424,2603,4967,8194,13663,22432,28169,47688,
%T A365300 65545,96615,146248,202507,266267,364834,450308,585328,773000,986339,
%U A365300 1162748,1472659,1993180,2275962,3012656,3552307,4590959,5404183,6601787,7893270,9340877
%N A365300 a(n) is the smallest nonnegative integer such that the sum of any four ordered terms a(k), k<=n (repetitions allowed), is unique.
%C A365300 This is the greedy B_4 sequence.
%H A365300 Chai Wah Wu, <a href="/A365300/b365300.txt">Table of n, a(n) for n = 1..50</a>
%H A365300 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 A365300 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 A365300 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 A365300 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 A365300 a(4) != 12 because 12+1+1+1 = 5+5+5+0.
%o A365300 (Python)
%o A365300 def GreedyBh(h, seed, stopat):
%o A365300     A = [set() for _ in range(h+1)]
%o A365300     A[1] = set(seed)    # A[i] will hold the i-fold sumset
%o A365300     for j in range(2,h+1): # {2,...,h}
%o A365300         for x in A[1]:
%o A365300             A[j].update([x+y for y in A[j-1]])
%o A365300     w = max(A[1])+1
%o A365300     while w <= stopat:
%o A365300         wgood = True
%o A365300         for k in range(1,h):
%o A365300             if wgood:
%o A365300                 for j in range(k+1,h+1):
%o A365300                     if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
%o A365300                         wgood = False
%o A365300         if wgood:
%o A365300             A[1].add(w)
%o A365300             for k in range(2,h+1): # update A[k]
%o A365300                 for j in range(1,k):
%o A365300                     A[k].update([(k-j)*w + x for x in A[j]])
%o A365300         w += 1
%o A365300         return A[1]
%o A365300 GreedyBh(4,[0],10000)
%o A365300 (Python)
%o A365300 from itertools import count, islice, combinations_with_replacement
%o A365300 def A365300_gen(): # generator of terms
%o A365300     aset, alist = set(), []
%o A365300     for k in count(0):
%o A365300         bset = set()
%o A365300         for d in combinations_with_replacement(alist+[k],3):
%o A365300             if (m:=sum(d)+k) in aset:
%o A365300                 break
%o A365300             bset.add(m)
%o A365300         else:
%o A365300             yield k
%o A365300             alist.append(k)
%o A365300             aset |= bset
%o A365300 A365300_list = list(islice(A365300_gen(),20)) # _Chai Wah Wu_, Sep 01 2023
%Y A365300 Row 4 of A365515.
%Y A365300 Cf. A025582, A051912, A365301, A365302, A365303, A365304, A365305.
%K A365300 nonn
%O A365300 1,3
%A A365300 _Kevin O'Bryant_, Aug 31 2023
%E A365300 a(27)-a(37) from _Chai Wah Wu_, Sep 01 2023