A373784 Number of n X n periodic matrices over GF(3).

1, 3, 73, 17083, 36899281, 723299706163, 127954611219059353, 203907700711025670584683, 2925406687944886208690683289761, 377768302929213133608351637048047649123, 439058106335101098121200963942043393433721555433, 4592676942963368381676178826582121303149079572759596838683

Offset: 0

Author

Geoffrey Critzer, Jun 18 2024

Keywords

• nonn

Comments

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

• Eric Weisstein's World of Mathematics, Periodic Matrix

Crossrefs

Cf. A348015, A053290.

Programs

Mathematica

nn = 11; B[n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n /. q -> 3;
e[x_] := Sum[x^n/B[n], {n, 0, nn}];f[x_] := Sum[(q - 1)^n x^n, {n, 0, nn}] /. q -> 3; Table[B[n], {n, 0, nn}] CoefficientList[Series[e[x] f[ x], {x, 0, nn}], x]

Formula

Sum_{n>=0}a_n*x^n/B(n) = e(x)*f(x) where B(n) = A053290(n)/2^n, e(x) = Sum{n>=0}x^n/B(n) and f(x) = Sum_{n>=0} (2x)^n.