cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A290516 Number of diagonalizable n X n matrices over GF(3).

Original entry on oeis.org

1, 3, 39, 2109, 417153, 346720179, 1233891662727, 17484682043488557, 1077565432934756756289, 290674711165255613845226787, 320439909778519092353160948081831, 1554385919734090411686737202215725913181, 33245671345010828575975932818988836416481765697
Offset: 0

Views

Author

Geoffrey Critzer, Aug 04 2017

Keywords

Crossrefs

Row sums of A296605.

Programs

  • Mathematica
    nn = 12; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
        QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^3, {z, 0, nn}], z]

Formula

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^3.

A364886 Number of n X n (-1, 1)-matrices which have only eigenvalues with strictly negative real part (which implies that the matrix has all nonzero eigenvalues).

Original entry on oeis.org

1, 2, 20, 640, 97824, 47545088
Offset: 1

Views

Author

Thomas Scheuerle, Aug 12 2023

Keywords

Comments

As this problem is symmetric with sign we can get the same numbers for strictly positive real parts.
All values for n > 1 are even, because a transposed matrix has the same spectrum of eigenvalues.
Matrices with determinant 0 are not counted.
Let M be such a matrix then the limit of ||exp(t*M)*y|| if t goes to infinity will be zero.
n = 5 is the first case where not all entries on the main diagonal are -1. 93984 matrices with 5 times -1 on the main diagonal and 5*768 with 4 times -1 on the main diagonal have only eigenvalues with strictly negative real part.
In the case n = 6, 43586048 matrices with 6 times -1 on the main diagonal, 6*656000 matrices with 5 times -1 on the main diagonal and 15*1536 matrices with 5 times -1 on the main diagonal have only eigenvalues with strictly negative real part.

Examples

			For n = 2 the matrices are:
.
    -1,  1
    -1, -1
.
    -1, -1
     1, -1.
		

Crossrefs

Showing 1-2 of 2 results.