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%I #32 Mar 01 2024 12:27:03
%S 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 32,56,72,31,8,1,2,9,44,109,154,125,45,9,1,2,10,58,155,367,369,219,66,
%U 10,1,2,11,74,257,669,927,857,376,81,11
%N Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.
%C 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 Chai Wah Wu, <a href="/A347570/b347570.txt">Table of n, a(n) for n = 1..241</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/B2-Sequence.html">B2 Sequence</a>.
%e Table begins:
%e n\k | 1 2 3 4 5 6 7 8
%e ----+------------------------------------------
%e 1 | 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 2 | 1, 2, 4, 8, 13, 21, 31, 45, ...
%e 3 | 1, 2, 5, 14, 33, 72, 125, 219, ...
%e 4 | 1, 2, 6, 22, 56, 154, 369, 857, ...
%e 5 | 1, 2, 7, 32, 109, 367, 927, 2287, ...
%e 6 | 1, 2, 8, 44, 155, 669, 2215, 6877, ...
%e 7 | 1, 2, 9, 58, 257, 1154, 4182, 14181, ...
%e 8 | 1, 2, 10, 74, 334, 1823, 8044, 28297, ...
%o (Python)
%o from itertools import count, islice, combinations_with_replacement
%o def A347570_gen(): # generator of terms
%o asets, alists, klist = [set()], [[]], [1]
%o while True:
%o for i in range(len(klist)-1,-1,-1):
%o kstart, alist, aset = klist[i], alists[i], asets[i]
%o for k in count(kstart):
%o bset = set()
%o for d in combinations_with_replacement(alist+[k],i):
%o if (m:=sum(d)+k) in aset:
%o break
%o bset.add(m)
%o else:
%o yield k
%o alists[i].append(k)
%o klist[i] = k+1
%o asets[i].update(bset)
%o break
%o klist.append(1)
%o asets.append(set())
%o alists.append([])
%o A347570_list = list(islice(A347570_gen(),30)) # _Chai Wah Wu_, Sep 06 2023
%Y Cf. A000027 (n=1), A005282 (n=2), A096772 (n=3), A014206 (k=4), A370754 (k=5).
%K nonn,tabl
%O 1,3
%A _Peter Kagey_, Sep 06 2021