A074033 Number of degree-n irreducible polynomials over GF(4) with trace 1 and subtrace 0.

1, 0, 1, 4, 15, 40, 144, 512, 1841, 6528, 23808, 87380, 322875, 1198080, 4473647, 16777216, 63164175, 238605640, 904200192, 3435973836, 13089461538, 49977753600, 191219367936, 733007751680, 2814750270420, 10825959997440, 41699995927744, 160842843834660

Offset

1,4

Comments

Let a = RootOf( x^2+x+1 ) and b = 1+a. Same as number of degree-n irreducible polynomials over GF(4) with trace a and subtrace 0. Same as number of degree-n irreducible polynomials over GF(4) with trace b and subtrace 0.

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

q = 4;
v[t_] := If[t === 0, q - 1, -1];
ddp[a_, n_] := q^(n-2) + q^Quotient[n-2, 2] {v[a], -1, v[1-a], 0}[[Mod[n, 4, 1]]];
h[n_, 1, a_] := 1/n Sum[MoebiusMu[d] ddp[Mod[a+(d-1)/2, 2], n/d], {d, Select[Divisors[n], OddQ]}];
Table[h[n, 1, 0], {n, 30}] (* this sequence *)
Table[h[n, 1, 1], {n, 30}] (* A074034 *)
(* Andrey Zabolotskiy, Dec 17 2020 *)

Crossrefs

Cf. A074031, A074032, A074034, A074035.

Sequence in context: A059140 A031164 A116600 * A093920 A342352 A011844

Adjacent sequences: A074030 A074031 A074032 * A074034 A074035 A074036

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 17 2020

Status

approved