A352940 The largest positive integer k such that binomial(k+1,2) <= binomial(n,2)^2.

3, 8, 13, 20, 29, 39, 50, 63, 77, 92, 109, 128, 147, 169, 191, 215, 241, 268, 296, 326, 357, 389, 423, 459, 495, 534, 573, 614, 657, 700, 746, 792, 840, 890, 941, 993, 1047, 1102, 1159, 1217, 1276, 1337, 1399, 1463, 1528, 1594, 1662, 1731, 1802, 1874, 1948

Offset: 3

Author

Nathaniel Johnston, May 06 2022

Keywords

• nonn

Comments

This sequence is bounded between floor((n-1)^2/sqrt(2) - 1) and (n-1)^2.

This sequence is the maximum dimension of a subspace of C^n * C^n (where * is the tensor/Kronecker product) that can be shown to be entangled by the first level of the hierarchy described in the linked Johnston-Lovitz-Vijayaraghavan paper.

Links

• N. Johnston, B. Lovitz, and A. Vijayaraghavan. Complete hierarchy of linear systems for certifying quantum entanglement of subspaces. Physical Review A, 106:062443, 2022.

Programs

Python

from math import isqrt
def A352940(n): return (isqrt(n**2*(n*(2*n-4)+2)+1)-1)//2 # Chai Wah Wu, May 07 2022

Formula

a(n) ~ (n-1)^2/sqrt(2).