A056989 Number of nonsingular n X n (-1,0,1)-matrices (over the reals).

1, 2, 48, 11808, 27947520, 609653621760, 119288919620689920

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• more

Comments

It would be nice to have an estimate for the asymptotic rate of growth.

Examples

			a(1) = 2: [1], [ -1].
a(2) = 48: There are 8 choices for the first column, u (say) and then the 2nd column can be anything except 0, u, -u, so 6 choices, giving a total of 8*6 = 48.
		

Links

• Minfeng Wang, C++ program to calculate A056989
• Eric Weisstein's World of Mathematics, Nonsingular Matrix
• Index entries for sequences related to binary matrices

Crossrefs

Cf. A055165, A053290, A056990, A055165, A002884, A046747, A057981, A060722.

Programs

Mathematica

(* A brute force solution up to n = 4 *) a[n_] := a[n] = (m = Array[x, {n, n}]; cnt = 0; iter = {#, -1, 1}& /@ Flatten[m]; Do[ If[ Det[m] != 0, cnt++], Evaluate[ Sequence @@ iter]]; cnt); Table[ Print[a[n]]; a[n], {n, 1, 4}] (* Jean-François Alcover, Oct 11 2012 *)

Formula

a(n) = A060722(n) - A057981(n). - Alois P. Heinz, Dec 02 2019

Extensions

a(4) from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000

Entry revised by N. J. A. Sloane, Jan 02 2007

a(5) from Giovanni Resta, Feb 20 2009

a(0)=1 prepended by Alois P. Heinz, Dec 02 2019

a(0)-a(5) confirmed and a(6) added by Minfeng Wang, May 01 2024