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 1-7 of 7 results.

A074024 Duplicate of A054661.

Original entry on oeis.org

1, 0, 5, 12, 51, 160, 585, 2016, 7280, 26112, 95325, 349180, 1290555, 4792320
Offset: 1

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Author

Keywords

A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

Original entry on oeis.org

1, 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).
Number of aperiodic necklaces with n beads of 4 colors. - Herbert Kociemba, Nov 25 2016

References

  • E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
  • M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Links

Crossrefs

Cf. A001037, A027376, A054719, A054660, A054661.
Column k=4 of A074650.

Programs

  • Maple
    A027377 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;
  • Mathematica
    a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* Jean-François Alcover, Nov 29 2011 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,4],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)
  • PARI
    a(n)=if(n,sumdiv(n,d,moebius(d)<<(2*n/d))/n,1) \\ Charles R Greathouse IV, Nov 29 2011

Formula

a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) = A054661(n) + 3 * A054660(n). - Andrey Zabolotskiy, Dec 17 2020
a(n) = 2 * (A054664(n) + A054660(n)). - Andrey Zabolotskiy, Dec 19 2020
a(n) = A054719(n)/n, n>0. - R. J. Mathar, Dec 16 2024

A054660 Number of monic irreducible polynomials over GF(4) of degree n with fixed nonzero trace.

Original entry on oeis.org

1, 2, 5, 16, 51, 170, 585, 2048, 7280, 26214, 95325, 349520, 1290555, 4793490, 17895679, 67108864, 252645135, 954437120, 3616814565, 13743895344, 52357696365, 199911205050, 764877654105, 2932031006720, 11258999068416
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of Lyndon words of length n with trace 1 over GF(4).
Let x = RootOf( z^2+z+1 ) and y = 1+x. Also number of Lyndon words of length n with trace x over GF(4). Also number of Lyndon words of length n with trace y over GF(4).
Also number of 4-ary Lyndon words (i.e., Lyndon words over Z_4) of length n with trace 1 (mod 4). Also the same with trace 3 (mod 4). - Andrey Zabolotskiy, Dec 19 2020

Examples

			a(3; y)=5 since the five Lyndon words over GF(4) of trace y and length 3 are { 00y, 01x, 0x1, 11y, xxy }; the five Lyndon words over Z_4 of trace 1 (mod 4) and length 3 are { 001, 023, 032, 113, 122 }.

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054661, etc.
Cf. A008683, A054661, A027377, A300628, A074033, A074034, A074035, A074448, A074449, A074450.
Cf. A074406, A074407, A074408, A074409.

Formula

From Seiichi Manyama, Mar 11 2018: (Start)
a(n) = A000048(2*n) = (1/(4*n)) * Sum_{odd d divides n} mu(d)*4^(n/d), where mu is the Möbius function A008683.
a(n+1) = A300628(n,n) for n >= 0. (End)
From Andrey Zabolotskiy, Dec 19 2020: (Start)
a(n) = A074033(n) + A074034(n) + 2 * A074035(n).
a(n) = A074448(n) + A074449(n) + 2 * A074450(n).
a(n) = A074406(n) + A074407(n) + A074408(n) + A074409(n). (End)

Extensions

More terms from James Sellers, Apr 19 2000

A054662 Number of monic irreducible polynomials over GF(5) with fixed nonzero trace.

Original entry on oeis.org

1, 2, 8, 30, 125, 516, 2232, 9750, 43400, 195250, 887784, 4068740, 18780048, 87191964, 406901000, 1907343750, 8975758272, 42385503300, 200773540296, 953674218750, 4541306267856, 21674415838068, 103660251783288
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 5-ary Lyndon words with trace 1 mod 5; trace 2 mod 5; trace 3 mod 5; trace 4 mod 5; also number of 5-ary Lyndon words of trace 1 over GF(5), trace 2 over GF(5); trace 3 over GF(5); trace 4 over GF(5).

Links

Crossrefs

Column 5 of A110540.
Cf. A000048, A051841, A046211, A046209, A054661, etc.

Programs

  • PARI
    a(n) = sumdiv(n, d, (gcd(d, 5)==1)*(moebius(d)*5^(n/d)))/(5*n); \\ Seiichi Manyama, May 29 2024

Formula

a(n) = 1/(5*n) * Sum_{d|n, gcd(d,5)=1} mu(d) * 5^(n/d). - Seiichi Manyama, May 29 2024

Extensions

More terms from James Sellers, Apr 19 2000

A074446 Number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace 0.

Original entry on oeis.org

1, 0, 2, 6, 15, 40, 153, 528, 1841, 6528, 23901, 87550, 322875, 1198080, 4474738, 16779264, 63164175, 238605640, 904213989, 3436000050, 13089461538, 49977753600, 191219550297, 733008101200, 2814750270420, 10825959997440, 41699998413248, 160842848628150, 621186162441675
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Examples

			a(3;0,0)=2 since the two 4-ary Lyndon words of trace 0, subtrace 0 and length 3 are { 123, 132 }.

Links

Crossrefs

Cf. A074447, A074448, A074449, A074450.
Cf. A054661, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021

A074447 Number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace 1.

Original entry on oeis.org

0, 0, 1, 2, 12, 40, 144, 496, 1813, 6528, 23808, 87210, 322560, 1198080, 4473647, 16775168, 63160320, 238605640, 904200192, 3435947622, 13089411609, 49977753600, 191219367936, 733007402160, 2814749599332, 10825959997440, 41699995927744, 160842839041170, 621186153185280
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Comments

Let x = RootOf( z^2+z+1 ) and y = 1+x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace y.

Examples

			a(4;0,1)=2 since the two 4-ary Lyndon words of trace 0, subtrace 1 and length 4 are { 0011, 11xx }, where x = RootOf( z^2+z+1 ).

Links

Crossrefs

Cf. A074446, A074448, A074449, A074450.
Cf. A054661, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021

A054663 Number of monic irreducible polynomials over GF(5) with zero trace.

Original entry on oeis.org

1, 2, 8, 30, 124, 516, 2232, 9750, 43400, 195248, 887784, 4068740, 18780048, 87191964, 406900992, 1907343750, 8975758272, 42385503300, 200773540296, 953674218720, 4541306267856, 21674415838068, 103660251783288
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 5-ary Lyndon words with trace 0 mod 5; also number of Lyndon words of trace 0 over GF(5). - Frank Ruskey and Nate Kube, Sep 11 2002

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054661, etc.

Extensions

More terms from James Sellers, Apr 19 2000
Showing 1-7 of 7 results.

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

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