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A347570 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.

%I A347570 #33 Feb 16 2025 08:34:02
%S A347570 1,1,2,1,2,3,1,2,4,4,1,2,5,8,5,1,2,6,14,13,6,1,2,7,22,33,21,7,1,2,8,
%T A347570 32,56,72,31,8,1,2,9,44,109,154,125,45,9,1,2,10,58,155,367,369,219,66,
%U A347570 10,1,2,11,74,257,669,927,857,376,81,11
%N A347570 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.
%C A347570 A B_n sequence is a sequence such that all sums a(x_1) + a(x_2) + ... + a(x_n) are distinct for 1 <= x_1 <= x_2 <= ... <= x_n.
%H A347570 Chai Wah Wu, <a href="/A347570/b347570.txt">Table of n, a(n) for n = 1..241</a>
%H A347570 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/B2-Sequence.html">B2 Sequence</a>.
%e A347570 Table begins:
%e A347570 n\k | 1  2   3   4    5     6     7      8
%e A347570 ----+------------------------------------------
%e A347570   1 | 1, 2,  3,  4,   5,    6,    7,     8, ...
%e A347570   2 | 1, 2,  4,  8,  13,   21,   31,    45, ...
%e A347570   3 | 1, 2,  5, 14,  33,   72,  125,   219, ...
%e A347570   4 | 1, 2,  6, 22,  56,  154,  369,   857, ...
%e A347570   5 | 1, 2,  7, 32, 109,  367,  927,  2287, ...
%e A347570   6 | 1, 2,  8, 44, 155,  669, 2215,  6877, ...
%e A347570   7 | 1, 2,  9, 58, 257, 1154, 4182, 14181, ...
%e A347570   8 | 1, 2, 10, 74, 334, 1823, 8044, 28297, ...
%o A347570 (Python)
%o A347570 from itertools import count, islice, combinations_with_replacement
%o A347570 def A347570_gen(): # generator of terms
%o A347570     asets, alists, klist = [set()], [[]], [1]
%o A347570     while True:
%o A347570         for i in range(len(klist)-1,-1,-1):
%o A347570             kstart, alist, aset = klist[i], alists[i], asets[i]
%o A347570             for k in count(kstart):
%o A347570                 bset = set()
%o A347570                 for d in combinations_with_replacement(alist+[k],i):
%o A347570                     if (m:=sum(d)+k) in aset:
%o A347570                         break
%o A347570                     bset.add(m)
%o A347570                 else:
%o A347570                     yield k
%o A347570                     alists[i].append(k)
%o A347570                     klist[i] = k+1
%o A347570                     asets[i].update(bset)
%o A347570                     break
%o A347570         klist.append(1)
%o A347570         asets.append(set())
%o A347570         alists.append([])
%o A347570 A347570_list = list(islice(A347570_gen(),30)) # _Chai Wah Wu_, Sep 06 2023
%Y A347570 Cf. A000027 (n=1), A005282 (n=2), A096772 (n=3), A014206 (k=4), A370754 (k=5).
%K A347570 nonn,tabl
%O A347570 1,3
%A A347570 _Peter Kagey_, Sep 06 2021