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A369818
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The sixth term of the greedy B_n set of natural numbers.
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3
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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
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OFFSET
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1,1
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COMMENTS
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{0, 1, n+1, n^2+n+1, A369817(n), a(n)} is the lexicographically first set of 6 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.
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LINKS
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FORMULA
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Conjectured that a(6n+i) is a quartic polynomial sequence with lead term (1/3)n^4 for each i in {1,2,3,5,6,10} in arxiv:2312.10910.
Proved that (1/8)*n^4 + (1/2)*n^3 <= a(n) <= 0.406671*n^4 + O(n^3) in arxiv:2312.10910.
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EXAMPLE
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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}.
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PROG
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(Python)
from itertools import count, combinations_with_replacement
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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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