A306838 Number of different values taken by the determinant of a real (-1,0,1) matrix of order n.

1, 3, 5, 9, 25, 67

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).

Links

Table of n, a(n) for n=0..5.

MathOverflow, Possible values of the determinant for matrices with elements {1,0,-1}

Steven E. Thornton, Properties of the Bohemian family of n x n matrices with population {-1, 0, 1}, Characteristic Polynomial Database.

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

Crossrefs

Number of matrices having maximum determinant is in A051753.

Sequence in context: A251413 A039774 A114001 * A171879 A171877 A339068

Adjacent sequences: A306835 A306836 A306837 * A306839 A306840 A306841

Keyword

nonn,more,hard

Author

Steven E. Thornton, Mar 12 2019

Extensions

Edited and expanded by Nathaniel Johnston, Apr 19 2022

Status

approved