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

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Keywords

• nice
• nonn

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

• 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

Crossrefs

Cf. A074031, A074032, A074034, A074035.

Programs

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 *)

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