A305503 Largest cardinality of subsets A of {0,1,...,n-1} with |A + A| > |A - A|.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40

Offset: 1

Author

Tanuj Mathur, Jun 03 2018

Keywords

• more
• hard
• nonn

Comments

All the possible 'A's are explicitly generated and sorted according to their cardinality.

Examples

			For n = 15, the subsets A of {0,1,...,n-1} with |A + A| > |A - A| are (0, 2, 3, 4, 7, 11, 12, 14); (0, 2, 3, 7, 10, 11, 12, 14); (0, 1, 2, 4, 5, 9, 12, 13, 14) and (0, 1, 2, 5, 9, 10, 12, 13, 14). So, the largest cardinality is 9.
		

Links

• P. V. Hegarty, Some explicit constructions of sets with more sums than differences, Acta Arith., 130 (2007), 61-77.
• Greg Martin and Kevin O'Bryant, Many sets have more sums than differences, arXiv:math/0608131 [math.NT], 2006-2007.

Crossrefs

Cf. A118544, A222807, A222808, A291514.

Programs

Python

import numpy as np
import itertools
def findsubsets(S, m):
    return itertools.combinations(S, m)
def mstd(a):
    a1 = set()
    a2 = set()
    for i in a:
        for j in a:
            a1.add(i + j)
            a2.add(i - j)
    return len(a1) > len(a2)
def a(n):
    ans = 0
    Nn = list(range(n))
    for k in range(1, n):
        if any(mstd(i) for i in findsubsets(Nn, k)):
            ans = k
    return ans

Formula

a(n) = n - 7 (conjectured) for all n > 15.

Conjectures from Colin Barker, Jun 01 2020: (Start)

G.f.: x^14*(9 - 9*x + x^2) / (1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>17.

(End)