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

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, 32, 56, 72, 31, 8, 1, 2, 9, 44, 109, 154, 125, 45, 9, 1, 2, 10, 58, 155, 367, 369, 219, 66, 10, 1, 2, 11, 74, 257, 669, 927, 857, 376, 81, 11

Offset: 1

Author

Peter Kagey, Sep 06 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			Table begins:
n\k | 1  2   3   4    5     6     7      8
----+------------------------------------------
  1 | 1, 2,  3,  4,   5,    6,    7,     8, ...
  2 | 1, 2,  4,  8,  13,   21,   31,    45, ...
  3 | 1, 2,  5, 14,  33,   72,  125,   219, ...
  4 | 1, 2,  6, 22,  56,  154,  369,   857, ...
  5 | 1, 2,  7, 32, 109,  367,  927,  2287, ...
  6 | 1, 2,  8, 44, 155,  669, 2215,  6877, ...
  7 | 1, 2,  9, 58, 257, 1154, 4182, 14181, ...
  8 | 1, 2, 10, 74, 334, 1823, 8044, 28297, ...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..241
• Eric Weisstein's World of Mathematics, B2 Sequence.

Crossrefs

Cf. A000027 (n=1), A005282 (n=2), A096772 (n=3), A014206 (k=4), A370754 (k=5).

Programs

Python

from itertools import count, islice, combinations_with_replacement
def A347570_gen(): # generator of terms
    asets, alists, klist = [set()], [[]], [1]
    while True:
        for i in range(len(klist)-1,-1,-1):
            kstart, alist, aset = klist[i], alists[i], asets[i]
            for k in count(kstart):
                bset = set()
                for d in combinations_with_replacement(alist+[k],i):
                    if (m:=sum(d)+k) in aset:
                        break
                    bset.add(m)
                else:
                    yield k
                    alists[i].append(k)
                    klist[i] = k+1
                    asets[i].update(bset)
                    break
        klist.append(1)
        asets.append(set())
        alists.append([])
A347570_list = list(islice(A347570_gen(),30)) # Chai Wah Wu, Sep 06 2023