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EXAMPLE
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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.
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MATHEMATICA
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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]
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