cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A054661 Number of monic irreducible polynomials over GF(4) with zero trace.

Original entry on oeis.org

1, 0, 5, 12, 51, 160, 585, 2016, 7280, 26112, 95325, 349180, 1290555, 4792320, 17895679, 67104768, 252645135, 954422560, 3616814565, 13743842916, 52357696365, 199911014400, 764877654105, 2932030307680, 11258999068416
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of Lyndon words of length n with trace 0 over GF(4).

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054660, A074031, A074032, A074446, A074447.

Formula

From Andrey Zabolotskiy, Dec 17 2020: (Start)
a(n) = A074031(n) + 3 * A074032(n).
a(n) = A074446(n) + 3 * A074447(n). (End)

Extensions

More terms from James Sellers, Apr 19 2000

A074032 Number of degree-n irreducible polynomials over GF(4) with trace 0 and subtrace 1.

Original entry on oeis.org

0, 0, 1, 4, 12, 40, 144, 512, 1813, 6528, 23808, 87380, 322560, 1198080, 4473647, 16777216, 63160320, 238605640, 904200192, 3435973836, 13089411609, 49977753600, 191219367936, 733007751680, 2814749599332, 10825959997440, 41699995927744, 160842843834660
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Keywords

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 0 and subtrace a. Same as number of degree-n irreducible polynomials over GF(4) with trace 0 and subtrace b.
Is this a duplicate of A074450? - R. J. Mathar, Dec 16 2020

Links

Crossrefs

Cf. A074031, A074033, A074034, A074035, A042979.
Cf. A074450.

Formula

a(n) = A042979(2*n)/2 [discovered by Sequence Machine]. - Andrey Zabolotskiy, Oct 09 2021

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

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

Original entry on oeis.org

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

Views

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Keywords

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

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

A074034 Number of degree-n irreducible polynomials over GF(4) with trace 1 and subtrace 1.

Original entry on oeis.org

0, 0, 2, 4, 12, 40, 153, 512, 1813, 6528, 23901, 87380, 322560, 1198080, 4474738, 16777216, 63160320, 238605640, 904213989, 3435973836, 13089411609, 49977753600, 191219550297, 733007751680, 2814749599332, 10825959997440, 41699998413248, 160842843834660
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Keywords

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

Links

Crossrefs

Cf. A074031, A074032, A074033, A074035.

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

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.

Original entry on oeis.org

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

Views

Author

Frank Ruskey and Nate Kube, Aug 26 2002

Keywords

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

Crossrefs

Cf. A074031, A074032, A074033, A074034.

Programs

  • 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 *)

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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