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A053846
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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)).
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8
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1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472
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OFFSET
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0,2
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COMMENTS
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Or, number of n X n invertible diagonalizable matrices over GF(3).
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REFERENCES
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V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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):
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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