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, 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).

Links

Table of n, a(n) for n=0..6.

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.

Minfeng Wang, C++ program to calculate A306838

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

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

a(6) from Minfeng Wang, May 31 2024

Status

approved