A074031 Number of degree-n irreducible polynomials over GF(4) with trace 0 and subtrace 0.

1, 0, 2, 0, 15, 40, 153, 480, 1841, 6528, 23901, 87040, 322875, 1198080, 4474738, 16773120, 63164175, 238605640, 904213989, 3435921408, 13089461538, 49977753600, 191219550297, 733007052640, 2814750270420, 10825959997440, 41699998413248, 160842834247680

Offset

1,3

Links

Table of n, a(n) for n=1..28.

E. N. Kuz'min, Irreducible polynomials over a finite field and an analogue of Gauss sums over a field of characteristic 2, Siberian Mathematical Journal, 32, 982-989 (1991).

Frank Ruskey, Number of irreducible polynomials over GF(4) with given trace and subtrace

Mathematica

k = 2; q = 2^k;
v[t_] := If[t === 0, q - 1, -1];
dd[a_, n_] := With[{m = Floor[(n + 1)/4]},
   q^(n - 2) + Switch[Mod[n, 4],
     2, 0,
     0, -v[a] q^((n - 2)/2) (-1)^(m k),
     _, v[a] q^((n - 3)/2) (-1)^(m k)
   ]];
h[n_, 0, a_] := 1/n Sum[MoebiusMu[d] (dd[a, n/d] - Boole[EvenQ[n]] q^(n/(2d)-1)), {d, Select[Divisors[n], OddQ]}];
Table[h[n, 0, 0], {n, 30}] (* this sequence *)
Table[h[n, 0, 1], {n, 30}] (* A074032 *)
(* Andrey Zabolotskiy, Dec 15 2020 *)

Crossrefs

Cf. A074032, A074033, A074034, A074035.

Sequence in context: A266169 A277928 A232423 * A086261 A111978 A146558

Adjacent sequences: A074028 A074029 A074030 * A074032 A074033 A074034

Keyword

nice,nonn

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Extensions

More terms from Ruskey's website added by Joerg Arndt, Jan 16 2011

Terms a(17) and beyond from Andrey Zabolotskiy, Dec 15 2020

Status

approved