cp's OEIS Frontend

A348015 Number of periodic n X n matrices over GF(2).

%I A348015 #56 Jan 04 2025 16:09:07
%S A348015 1,2,13,365,43801,21725297,43798198753,355991759464385,
%T A348015 11619571028917526401,1520025803718875133673217,
%U A348015 796153035368657542014822907393,1668838669721233396228446711227874305,13995815633937307151473642050515241531340801
%N A348015 Number of periodic n X n matrices over GF(2).
%C A348015 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.
%H A348015 Alois P. Heinz, <a href="/A348015/b348015.txt">Table of n, a(n) for n = 0..57</a>
%H A348015 Geoffrey Critzer, <a href="/A348015/a348015_1.pdf">Enumeration of matrices over finite fields</a>
%H A348015 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PeriodicMatrix.html">Periodic Matrix</a>
%F A348015 a(n) = Sum_{d=0...n} A002884(n)/A002884(n-d). - _Geoffrey Critzer_, Oct 30 2021
%F A348015 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
%F A348015 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
%F A348015 Sum_{n>=0} a(n)*x^n/B(n) = e(x)*g(x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - _Geoffrey Critzer_, Jan 03 2025
%e A348015 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.
%p A348015 b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
%p A348015 a:= n-> add(b(n)/b(n-k), k=0..n):
%p A348015 seq(a(n), n=0..14);  # _Alois P. Heinz_, Oct 30 2021
%t A348015 nn=12; q = 2; b[p_, i_] := Count[p, i];
%t A348015 s[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];
%t A348015 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}];
%t A348015 g1[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[1]]];
%t A348015 g2[u_, v_, deg_] :=Total[Map[v u^(deg Total[#])/aut[deg, #] &, l[nn]]];
%t A348015 Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
%t A348015   Series[g1[u, 1, 1] g2[u, 1, 1] Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}] , {u, 0, nn}], u]
%Y A348015 Cf. A002884.
%Y A348015 Row sums of A348622.
%K A348015 nonn
%O A348015 0,2
%A A348015 _Geoffrey Critzer_, Sep 24 2021
%E A348015 Data terms for n >= 3 corrected by _Geoffrey Critzer_, Oct 30 2021
%E A348015 Title improved by _Geoffrey Critzer_, Sep 16 2022