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A053846 Number of n X n matrices over GF(3) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,3)). 8
1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, number of n X n invertible diagonalizable matrices over GF(3).

REFERENCES

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..60

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

FORMULA

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - Geoffrey Critzer, Aug 05 2017

EXAMPLE

a(2) = 14 because we have: {{0, 1}, {1, 0}}, {{0, 2}, {2, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {0,2}}, {{1, 0}, {1, 2}}, {{1, 0}, {2, 2}}, {{1, 1}, {0, 2}}, {{1,2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 0}, {1,1}}, {{2, 0}, {2, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}. -  Geoffrey Critzer, Aug 05 2017

MAPLE

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,

      `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))

    end:

a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):

seq(a(n), n=0..15);  # Alois P. Heinz, Aug 06 2017

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]^2, {z, 0, nn}], z] /* Geoffrey Critzer, Aug 05 2017

PROG

(PARI) a(n)={my(v=[1]); for(n=1, n, v=vector(#v+1, k, if(k>1, v[k-1]) + if(k<=#v, 3^(k-1)*v[k]))); sum(k=0, n, 3^(k*(n-k))*v[k+1])} \\ Andrew Howroyd, Mar 02 2018

(Python)

from sympy.core.cache import cacheit

@cacheit

def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else T(n - 1, k - 1) + 3**k*T(n - 1, k)

def a(n): return sum([3**(k*(n - k))*T(n, k) for k in xrange(n + 1)])

print map(a, xrange(15)) # Indranil Ghosh, Aug 06 2017, after Maple code

CROSSREFS

Cf. A053722, A053290, A290516.

Sequence in context: A048163 A093548 A052215 * A053855 A219344 A152476

Adjacent sequences:  A053843 A053844 A053845 * A053847 A053848 A053849

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Mar 28 2000

EXTENSIONS

More terms from Geoffrey Critzer, Aug 05 2017

STATUS

approved

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Last modified March 23 05:36 EDT 2019. Contains 321422 sequences. (Running on oeis4.)