login
A306838
Number of different values taken by the determinant of a real (-1,0,1) matrix of order n.
1
1, 3, 5, 9, 25, 67, 233
OFFSET
0,2
COMMENTS
Every term in this sequence is odd, since 0 is a possible determinant, and if d is a possible determinant then so is -d.
a(n) >= 1 + 2^n, since every integer determinant between -2^(n-1) and 2^(n-1) is possible (see MathOverflow link).
EXAMPLE
For n = 2, the possible determinants of a 2x2 matrix with entries from {-1,0,1} are -2, -1, 0, 1, and 2. Since there are 5 numbers in this list, a(2) = 5.
The possible nonnegative determinants for small values of n are as follows (all the negatives of these numbers are also possible determinants):
n = 1: 0, 1
n = 2: 0, 1, 2
n = 3: 0, 1, 2, 3, 4
n = 4: 0 through 10, 12, 16
n = 5: 0 through 28, 30, 32, 36, 40, 48
n = 6: 0 through 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 120, 125, 128, 130, 132, 136, 144, 160
CROSSREFS
Number of matrices having maximum determinant is in A051753.
Sequence in context: A251413 A039774 A114001 * A171879 A171877 A339068
KEYWORD
nonn,more,hard
AUTHOR
Steven E. Thornton, Mar 12 2019
EXTENSIONS
Edited and expanded by Nathaniel Johnston, Apr 19 2022
a(6) from Minfeng Wang, May 31 2024
STATUS
approved