A378126 -id:A378126 - cp's OEIS frontend

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

1, 1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16, 18, 20, 23, 25, 28, 30, 33, 35, 38, 41, 44, 47, 50, 53, 56, 60, 63, 67, 70, 74, 77, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 129, 134, 138, 143, 147, 152, 156, 161, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220

Offset: 1

Jimin Park, Nov 24 2024

nonn

For (n, n), there is at least one maximal partition P that's symmetric: (x, y) in P <=> (y, x) in P. This can be proven by manipulating integer sequences c(i) (i >= 1) such that 0 <= c(i) <= i+1 for all i and Sum_{i > 0} i*c(i) = 2n, which correspond to partitions P of (n, n) with size |P| = Sum_{i > 0} c(i), where c(i) is equal to number of (x, y) in P such that x+y = i.

			For n = 8, a(n) = 9, as (9, 9) can be expressed as the sum (0, 1) + (0, 2) + (0, 3) + (1, 0) + (2, 0) + (3, 0) + (1, 2) + (2, 1), but the longest sum for (8, 8) has 7 pairs.

Maximal size among partitions considered by A054242 and A201377.
Minimal x such that A378126(x, x) >= n.
Cf. A086435.

import math
def A378379(n: int) -> int:
  l = (math.isqrt(1+8*n)-1)//2 # l = A003056(n), min. possible largest pair norm
  r = n - (l-1)*(l+2)//2 # r = n - A000096(l-1), number of pairs with norm l
  return ((l-1)*l*(l+1)//3 + l*r + 1)//2 # ceil((A007290(l+1) + l*r) / 2)

a(n*(n+3)/2) = n*(n+1)*(n+2)/6.

cp's OEIS Frontend

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

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