A369818 -id:A369818 - cp's OEIS frontend

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

Chai Wah Wu, Sep 07 2023

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.

			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, ...

Chai Wah Wu, Table of n, a(n) for n = 1..242
Eric Weisstein's World of Mathematics, B2 Sequence.

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.

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))

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

A369819 The seventh term of the greedy B_n set of natural numbers.

Original entry on oeis.org

6, 30, 124, 368, 926, 2214, 4181, 8043, 13818, 23614, 34825, 54011, 84026, 109870, 156474, 217790, 304910, 376260, 510220, 667130, 794873, 1008048, 1302947, 1629264, 1916949, 2361150, 2859694, 3467661, 3989744, 4779270, 5479857, 6449983, 7575912

Offset: 1

Kevin O'Bryant, Feb 03 2024

Proved in arXiv:2312.10910 that a(n) <= 0.382978*n^5 + O(n^4).

			a(2) = 30, as all 28 nonincreasing sums from {0,1,3,7,12,20,30}, namely 0+0 < 0+1 < 1+1 < ... < 7+20 < 0+30 < 1+30 < 12+20 <3+30 < 7+30 < 20+20 < 12+30 < 20+30 < 30+30, are distinct, and all other 7-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20,30}.

M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.

Column 7 of A365515.

# uses Python code from A369818
from itertools import count, combinations_with_replacement
def A369819(n):
    alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1),A369818(n)]
    aset = set(sum(d) for d in combinations_with_replacement(alist,n))
    blist = []
    for i in range(n):
        blist.append(set(sum(d) for d in combinations_with_replacement(alist,i)))
    for k in count(alist[-1]+1):
        for i in range(n):
            if any((n-i)*k+d in aset for d in blist[i]):
                break
        else:
            return k # Chai Wah Wu, Feb 28 2024

cp's OEIS Frontend

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

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