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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V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.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
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 range(n + 1))
print([a(n) for n in range(15)]) # Indranil Ghosh, Aug 06 2017, after Maple code
CROSSREFS
Sequence in context: A048163 A093548 A052215 * A053855 A219344 A343441
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 19 07:41 EDT 2024. Contains 370958 sequences. (Running on oeis4.)