A365515
Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence starting from 0.
Original entry on oeis.org
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
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, ...
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.
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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))
A369818
The sixth term of the greedy B_n set of natural numbers.
Original entry on oeis.org
5, 20, 71, 153, 366, 668, 1153, 1822, 3119, 4448, 6348, 8559, 11565, 14976, 21023, 26220, 33066, 40306, 49601, 59354, 76031, 89248, 106008, 122909, 143989, 165196, 200759, 227660, 261030, 293736, 333825, 373110, 438191, 485952, 544356, 600523, 668573, 734072, 841679, 918988, 1012578, 1101374, 1208065, 1309426, 1474943, 1592000, 1732656
Offset: 1
a(2) = 20, as all 21 nonincreasing sums from {0,1,3,7,12,20}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 0+20 < 1+20 < 3+20 < 12+12 < 7+20 < 12+20 < 20+20, are distinct, and all other 6-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20}.
-
from itertools import count, combinations_with_replacement
def A369818(n):
alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1)]
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(max(alist[-1]+1,(n**3>>1)*(1+(n>>2)))):
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
A370754
a(n) = 2 + n^2*floor((n+3)/2) + floor(3*n/2).
Original entry on oeis.org
5, 13, 33, 56, 109, 155, 257, 334, 501, 617, 865, 1028, 1373, 1591, 2049, 2330, 2917, 3269, 4001, 4432, 5325, 5843, 6913, 7526, 8789, 9505, 10977, 11804, 13501, 14447, 16385, 17458, 19653, 20861, 23329, 24680, 27437, 28939, 32001, 33662, 37045, 38873, 42593, 44596
Offset: 1
- Paolo Xausa, Table of n, a(n) for n = 1..10000
- 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.
- Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
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A370754[n_] := 2 + n^2*Floor[(n+3)/2] + Floor[3*n/2]; Array[A370754, 50] (* or *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {5, 13, 33, 56, 109, 155, 257}, 50] (* Paolo Xausa, Mar 08 2024 *)
Showing 1-3 of 3 results.
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