A231074
The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in nondecreasing order.
Original entry on oeis.org
1, 1, 1, 1, 2, 12, 244
Offset: 0
Let a < b < c < d. There are two possible ways to arrange the sums in nondecreasing order:
1) a+b <= a+c <= a+d <= b+c <= b+d <= c+d, (for instance, a = 1, b = 3, c = 4, d = 5);
2) a+b <= a+c <= b+c <= a+d <= b+d <= c+d, (for instance, a = 1, b = 2, c = 3, d = 5).
Hence a(4) = 2.
A231085
The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in increasing order provided that all sums are different.
Original entry on oeis.org
1, 1, 1, 1, 2, 12, 168, 4680
Offset: 0
Let a < b < c < d. There are two possible ways to arrange the sums in increasing order:
1) a+b < a+c < a+d < b+c < b+d < c+d, (for instance, a = 1, b = 3, c = 4, d = 5);
2) a+b < a+c < b+c < a+d < b+d < c+d, (for instance, a = 1, b = 2, c = 3, d = 5).
Hence a(4) = 2.
A259762
Smallest integer k_1 such that there exist n positive integers k_1 > k_2 > ... > k_n having the property that k_j * k_n > k_(j+1)^2 for j=1..n-1.
Original entry on oeis.org
1, 2, 5, 13, 29, 68, 145, 307, 636, 1312, 2659, 5404, 10892, 21937, 44039, 88416, 177136, 354965, 710576, 1422447, 2846284, 5695248, 11393091, 22791749, 45588844, 91188435, 182387991, 364797722, 729617037, 1459278556, 2918600648, 5837288849, 11674666710, 23349509456, 46699194308, 93398744563
Offset: 1
The positive integer triple (k_1,k_2,k_3) = (5,2,1) yields pairwise products in the required ordering; i.e.,
k_1 * k_1 > k_1 * k_2 > k_1 * k_3
> k_2 * k_2 > k_2 * k_3
> k_3 * k_3
becomes
5*5 > 5*2 > 5*1
> 2*2 > 2*1
> 1*1
i.e.,
25 > 10 > 5
> 4 > 2
> 1
which verifies that the requirement is satisfied. The triple (5,3,2) also satisfies the requirement, but there exists no such triple with k_1 < 5, so a(3) = 5.
Similarly, there exist quadruples that meet the requirement (the ones whose largest member is 13 are (13,5,3,2), (13,6,4,3), (13,7,5,4), and (13,8,6,5)), but there is no such quadruple with k_1 < 13, so a(4) = 13.
Of the quintuples that meet the requirement, (29,17,13,11,10) is the only one with k_1 = 29, and there is no such quintuple with k_1 < 29, so a(5) = 29.
Showing 1-3 of 3 results.
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