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
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.
Programs
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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)
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