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 A348015 Number of periodic n X n matrices over GF(2). 3
 1, 2, 13, 365, 43801, 21725297, 43798198753, 355991759464385, 11619571028917526401, 1520025803718875133673217, 796153035368657542014822907393, 1668838669721233396228446711227874305, 13995815633937307151473642050515241531340801 (list; graph; refs; listen; history; text; internal format)
 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 a(n) = Sum_{d=0...n} A002884(n)/A002884(n-d). - Geoffrey Critzer, Oct 30 2021 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 Cf. A002884, A348622. Sequence in context: A355730 A075620 A336188 * A131525 A082751 A120935 Adjacent sequences: A348012 A348013 A348014 * A348016 A348017 A348018 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

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Last modified September 28 12:43 EDT 2023. Contains 365735 sequences. (Running on oeis4.)