A365515 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence starting from 0.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 7, 4, 0, 1, 5, 13, 12, 5, 0, 1, 6, 21, 32, 20, 6, 0, 1, 7, 31, 55, 71, 30, 7, 0, 1, 8, 43, 108, 153, 124, 44, 8, 0, 1, 9, 57, 154, 366, 368, 218, 65, 9, 0, 1, 10, 73, 256, 668, 926, 856, 375, 80, 10, 0, 1, 11, 91, 333, 1153, 2214, 2286, 1424, 572, 96, 11

Offset

1,6

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. Analogous to A347570 except that here the B_n sequences start from a(1) = 0.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..242

Eric Weisstein's World of Mathematics, B2 Sequence.

Formula

a(n) = A347570(n)-1.

Example

Table begins:
n\k | 1  2   3   4    5     6      7      8       9
----+---------------------------------------------------
  1 | 0, 1,  2,  3,   4,    5,     6,     7,      8, ...
  2 | 0, 1,  3,  7,  12,   20,    30,    44,     65, ...
  3 | 0, 1,  4, 13,  32,   71,   124,   218,    375, ...
  4 | 0, 1,  5, 21,  55,  153,   368,   856,   1424, ...
  5 | 0, 1,  6, 31, 108,  366,   926,  2286,   5733, ...
  6 | 0, 1,  7, 43, 154,  668,  2214,  6876,  16864, ...
  7 | 0, 1,  8, 57, 256, 1153,  4181, 14180,  47381, ...
  8 | 0, 1,  9, 73, 333, 1822,  8043, 28296, 102042, ...
  9 | 0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, ...

Prog

(Python)
from itertools import count, islice, combinations_with_replacement
def A365515_gen(): # generator of terms
    asets, alists, klist = [set()], [[]], [0]
    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(0)
        asets.append(set())
        alists.append([])
A365515_list = list(islice(A365515_gen(), 30))

Crossrefs

Cf. A001477 (n=1), A025582 (n=2), A051912 (n=3), A365300 (n=4), A365301 (n=5), A365302 (n=6), A365303 (n=7), A365304 (n=8), A365305 (n=9), A002061 (k=4), A369817 (k=5), A369818 (k=6), A369819 (k=7), A347570.

Sequence in context: A198321 A325003 A166278 * A316269 A242379 A103438

Adjacent sequences: A365512 A365513 A365514 * A365516 A365517 A365518

Keyword

nonn,tabl

Author

Chai Wah Wu, Sep 07 2023

Status

approved