A074035 Let a = RootOf( x^2+x+1 ) and b = 1+a. Number of degree-n irreducible polynomials over GF(4) with trace 1 and subtrace a.

0, 1, 1, 4, 12, 45, 144, 512, 1813, 6579, 23808, 87380, 322560, 1198665, 4473647, 16777216, 63160320, 238612920, 904200192, 3435973836, 13089411609, 49977848925, 191219367936, 733007751680, 2814749599332, 10825961287995, 41699995927744, 160842843834660

Offset

1,4

Comments

Same as number of degree-n irreducible polynomials over GF(4) with trace 1 and subtrace b. Same as number of degree-n irreducible polynomials over GF(4) with trace a and subtrace 1. Same as number of degree-n irreducible polynomials over GF(4) with trace a and subtrace a. Same as number of degree-n irreducible polynomials over GF(4) with trace b and subtrace 1. Same as number of degree-n irreducible polynomials over GF(4) with trace b and subtrace b.

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, The Combinatorial Object Server.

Mathematica

q = 4;
ddpx[n_] := q^(n-2) + q^Quotient[n-2, 2] {-1, 1, -1, 0}[[Mod[n, 4, 1]]];
h1x[n_] :=  1/n Sum[MoebiusMu[d] ddpx[n/d], {d, Select[Divisors[n], OddQ]}];
Table[h1x[n], {n, 30}]
(* Andrey Zabolotskiy, Dec 17 2020 *)

Crossrefs

Cf. A074031, A074032, A074033, A074034.

Sequence in context: A098543 A028407 A105344 * A149366 A370540 A149367

Adjacent sequences: A074032 A074033 A074034 * A074036 A074037 A074038

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