A378379 - OEIS

%I #16 Dec 12 2024 23:08:39

%S 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,

%T 53,56,60,63,67,70,74,77,81,84,88,92,96,100,104,108,112,116,120,125,

%U 129,134,138,143,147,152,156,161,165,170,175,180,185,190,195,200,205,210,215,220

%N 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).

%C 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.

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

%e 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.

%o (Python)

%o import math

%o def A378379(n: int) -> int:

%o   l = (math.isqrt(1+8*n)-1)//2 # l = A003056(n), min. possible largest pair norm

%o   r = n - (l-1)*(l+2)//2 # r = n - A000096(l-1), number of pairs with norm l

%o   return ((l-1)*l*(l+1)//3 + l*r + 1)//2 # ceil((A007290(l+1) + l*r) / 2)

%Y Maximal size among partitions considered by A054242 and A201377.

%Y Minimal x such that A378126(x, x) >= n.

%Y Cf. A086435.

%K nonn

%O 1,3

%A _Jimin Park_, Nov 24 2024