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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)).

%I A053846 #48 Aug 04 2025 03:37:22
%S A053846 1,2,14,236,12692,1783784,811523288,995733306992,3988947598331024,
%T A053846 43581058503809001248,1559669026899267564563936,
%U A053846 152805492791495918971070907584,49094725258525117931062810300451648,43237014297639482582550110281347475757696,124920254287369111633119733942816364074145497472
%N 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)).
%C A053846 Or, number of n X n invertible diagonalizable matrices over GF(3).
%D A053846 Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%H A053846 Andrew Howroyd, <a href="/A053846/b053846.txt">Table of n, a(n) for n = 0..60</a>
%H A053846 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A053846 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A053846 a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - _Geoffrey Critzer_, Aug 05 2017
%e A053846 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
%p A053846 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A053846       `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))
%p A053846     end:
%p A053846 a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):
%p A053846 seq(a(n), n=0..15);  # _Alois P. Heinz_, Aug 06 2017
%t A053846 nn = 14; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
%t A053846 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 *)
%o A053846 (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
%o A053846 (Python)
%o A053846 from sympy.core.cache import cacheit
%o A053846 @cacheit
%o A053846 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)
%o A053846 def a(n): return sum(3**(k*(n - k))*T(n, k) for k in range(n + 1))
%o A053846 print([a(n) for n in range(15)]) # _Indranil Ghosh_, Aug 06 2017, after Maple code
%Y A053846 Cf. A053722, A053290, A290516.
%Y A053846 Row sums of A378666.
%K A053846 nonn
%O A053846 0,2
%A A053846 _Vladeta Jovovic_, Mar 28 2000
%E A053846 More terms from _Geoffrey Critzer_, Aug 05 2017