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

Author

Steven E. Thornton, Mar 12 2019

Keywords

• nonn
• more
• hard

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).

Examples

			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
		

Links

• 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

Crossrefs

Number of matrices having maximum determinant is in A051753.

Extensions

Edited and expanded by Nathaniel Johnston, Apr 19 2022

a(6) from Minfeng Wang, May 31 2024