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Number of degree-n irreducible polynomials over GF(4) with trace 1 and subtrace 0.
5

%I #24 Dec 17 2020 14:23:40

%S 1,0,1,4,15,40,144,512,1841,6528,23808,87380,322875,1198080,4473647,

%T 16777216,63164175,238605640,904200192,3435973836,13089461538,

%U 49977753600,191219367936,733007751680,2814750270420,10825959997440,41699995927744,160842843834660

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

%C 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.

%H E. N. Kuz'min, <a href="https://doi.org/10.1007/BF00971203">Irreducible polynomials over a finite field and an analogue of Gauss sums over a field of characteristic 2</a>, Siberian Mathematical Journal, 32, 982-989 (1991).

%H Frank Ruskey, <a href="http://combos.org/TSpoly4">Number of irreducible polynomials over GF(4) with given trace and subtrace</a>

%t q = 4;

%t v[t_] := If[t === 0, q - 1, -1];

%t ddp[a_, n_] := q^(n-2) + q^Quotient[n-2, 2] {v[a], -1, v[1-a], 0}[[Mod[n, 4, 1]]];

%t h[n_, 1, a_] := 1/n Sum[MoebiusMu[d] ddp[Mod[a+(d-1)/2, 2], n/d], {d, Select[Divisors[n], OddQ]}];

%t Table[h[n, 1, 0], {n, 30}] (* this sequence *)

%t Table[h[n, 1, 1], {n, 30}] (* A074034 *)

%t (* _Andrey Zabolotskiy_, Dec 17 2020 *)

%Y Cf. A074031, A074032, A074034, A074035.

%K nice,nonn

%O 1,4

%A _Frank Ruskey_ and Nate Kube, Aug 26 2002

%E More terms from Ruskey's website added by _Joerg Arndt_, Jan 16 2011

%E Terms a(17) and beyond from _Andrey Zabolotskiy_, Dec 17 2020