OFFSET
0,2
COMMENTS
Here, T is a periodic matrix if T = T^k for some k > 1. T is periodic iff image(T) = image(T^2) iff x^2 does not divide the minimal polynomial of T.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..57
Geoffrey Critzer, Enumeration of matrices over finite fields
Eric Weisstein's World of Mathematics, Periodic Matrix
FORMULA
Sum_{n>=0} a(n)u^n/A002884(n) = E(u)/(1-u) where E(u) = Sum_{n>=0} u^n/A002884(n). - Geoffrey Critzer, Oct 30 2021
Limit_{n->infinity} a(n)/2^(n^2) = (Product_{r>=1} (1-1/2^r)) * Sum_{n>=0} 1/A002884(n) = 0.62744086981206237307... . - Geoffrey Critzer, Oct 30 2021
EXAMPLE
a(2) = 13 because there are 16 2 X 2 matrices over GF(2) and all are recurrent except for {{0,0},{1,0}}, {{0,1},{0,0}}, and {{1,1},{1,1}}. 16-3 = 13.
MAPLE
b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
a:= n-> add(b(n)/b(n-k), k=0..n):
seq(a(n), n=0..14); # Alois P. Heinz, Oct 30 2021
MATHEMATICA
nn=12; q = 2; b[p_, i_] := Count[p, i];
s[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];
aut[deg_, p_] := Product[Product[q^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] =Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1, nn}]; l[greatestpart_] := Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
g1[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[nn]]];
Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
Series[g1[u, 1, 1] g2[u, 1, 1] Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}] , {u, 0, nn}], u]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 24 2021
EXTENSIONS
Data terms for n >= 3 corrected by Geoffrey Critzer, Oct 30 2021
Title improved by Geoffrey Critzer, Sep 16 2022
STATUS
approved