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 1-6 of 6 results.

A046209 Number of ternary Lyndon words whose digits sum to 0 mod 3; also number of trace 0 irreducible polynomials over GF(3).

Original entry on oeis.org

1, 1, 2, 6, 16, 38, 104, 270, 726, 1960, 5368, 14736, 40880, 113828, 318848, 896670, 2532160, 7174050, 20390552, 58112088, 166037248, 475467916, 1364393896, 3922624800, 11297181456, 32588003000, 94143178098, 272342710380, 788854912240, 2287679084096, 6641649422408, 19302293185470
Offset: 1

Views

Author

Frank Ruskey, Dec 13 1999

Keywords

Comments

Also number of ternary Lyndon words of trace 0 over GF(3).

Examples

			a(4) = 6 = |{ 0012, 0021, 0111, 0102, 0222, 1122 }|.

Links

Crossrefs

Cf. A046211.

Programs

  • Mathematica
    a[n_] := 1/(3n) DivisorSum[n, GCD[#, 3]*MoebiusMu[#]*3^(n/#)&]; Array[a, 32] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
  • PARI
    a(n) = 1/(3*n) * sumdiv(n, d, gcd(d, 3)*moebius(d)*3^(n/d) ); /* Joerg Arndt, Aug 17 2012 */

Formula

a(n) = 1/(3*n) * sum(d divides n, gcd(d, 3)*mu(d)*3^(n/d) ).
a(n) = A053548(n) + A053560(n) + A053561(n). - R. J. Mathar, Oct 21 2021

A053548 Number of ternary Lyndon words of length n with trace 0 and subtrace 0 over GF(3).

Original entry on oeis.org

1, 0, 0, 2, 4, 9, 32, 90, 240, 654, 1804, 4950, 13664, 37944, 106272, 298890, 843796, 2390595, 6796160, 19370696, 55345680, 158489298, 454803100, 1307556162, 3765741324, 10862667648, 31381058880, 90780903460, 262951527460
Offset: 1

Views

Author

Frank Ruskey, Jan 16 2000

Keywords

Comments

Trace is sum of digits, subtrace is sum of products of pairs of digits. [3|n] above is "Iversonian convention", 1 if 3|n, 0 otherwise.

Examples

			a(4) = 2 = |{ 0111, 0222 }|

Links

Crossrefs

Cf. A053560, A053561, A053562, A053563, A053564.

Formula

a(n) = (1/n) * Sum_{d divides n, d==1, 2(3)} mu(d) * (M(n/d, 0, 0)-[3*d divides n] * 3^{n/(3*d)}), where M(n, t, s) = Sum_{i+j+k=n, j=t(3), k=s(3)} n!/(i!*j!*k!). [Corrected by Sean A. Irvine, Dec 27 2021]

A053561 Number of ternary Lyndon words of length n with trace 0 and subtrace 2 over GF(3).

Original entry on oeis.org

0, 1, 2, 3, 6, 15, 36, 87, 234, 645, 1782, 4893, 13608, 37994, 106434, 299025, 844182, 2391723, 6797196, 19369708, 55342972, 158486625, 454795398, 1307534319, 3765720066, 10862688116, 31381118658, 90780960426, 262951692390
Offset: 1

Views

Author

Frank Ruskey, Jan 17 2000

Keywords

Examples

			a(4) = 3 = |{ 0012, 0021, 0102 }|

Links

Crossrefs

Cf. A053548, A053560, A053562, A053563, A053564.

Formula

(1/n) Sum mu(d) M(n/d, 1, 1); d divides n, d=1(3) + (1/n) Sum mu(d) M(n/d, 2, 2); d divides n, d=2(3) where M(n, t, s) = Sum n!/(i!j!k!); i+j+k=n, j=t(3), k=s(3)

A053562 Number of ternary Lyndon words of length n with trace 1 and subtrace 0 over GF(3). Same as the number of ternary Lyndon words of length n with trace 2 and subtrace 0 over GF(3).

Original entry on oeis.org

1, 1, 1, 2, 6, 13, 32, 87, 243, 654, 1782, 4914, 13664, 37994, 106288, 298890, 844182, 2391363, 6796160, 19369708, 55345784, 158489298, 454795398, 1307541690, 3765741324, 10862688116, 31381059609, 90780903460, 262951692390
Offset: 1

Views

Author

Frank Ruskey, Jan 17 2000

Keywords

Examples

			a(4) = 3 = |{ 0001, 1222 }|

Links

Crossrefs

Cf. A053548, A053560, A053561, A053563, A053564.

Formula

(1/n) Sum mu(d) M(n/d, 0, 1); d|n, d=1(3) + (1/n) Sum mu(d) M(n/d, 0, 2); d|n, d=2(3) where M(n, t, s) = Sum n!/(i!j!k!); i+j+k=n, j=t(3), k=s(3).

A053563 Number of ternary Lyndon words of length n with trace 1 and subtrace 1 over GF(3). Same as the number of ternary Lyndon words of length n with trace 2 and subtrace 1 over GF(3).

Original entry on oeis.org

0, 0, 1, 1, 4, 13, 36, 90, 243, 661, 1804, 4914, 13608, 37944, 106288, 298755, 843796, 2391363, 6797196, 19370696, 55345784, 158491993, 454803100, 1307541690, 3765720066, 10862667648, 31381059609, 90780846494, 262951527460
Offset: 1

Views

Author

Frank Ruskey, Jan 17 2000

Keywords

Examples

			a(4) = 1 = |{ 0022 }|

Links

Crossrefs

Cf. A053548, A053560, A053561, A053562, A053564.

Formula

(1/n) Sum mu(d) M(n/d, 0, 2); d|n, d=1(3) + (1/n) Sum mu(d) M(n/d, 0, 1); d|n, d=2(3) where M(n, t, s) = Sum n!/(i!j!k!); i+j+k=n, j=t(3), k=s(3).

A053564 Number of ternary Lyndon words of length n with trace 1 and subtrace 2 over GF(3). Same as the number of ternary Lyndon words of length n with trace 2 and subtrace 2 over GF(3).

Original entry on oeis.org

0, 0, 1, 3, 6, 13, 36, 93, 243, 645, 1782, 4914, 13608, 37890, 106288, 299025, 844182, 2391363, 6797196, 19371684, 55345784, 158486625, 454795398, 1307541690, 3765720066, 10862647236, 31381059609, 90780960426, 262951692390
Offset: 1

Views

Author

Frank Ruskey, Jan 17 2000

Keywords

Examples

			a(4) = 3 = |{ 0112, 0121, 0211 }|

Links

Crossrefs

Cf. A053548, A053560, A053561, A053562, A053563.

Formula

(1/n) Sum mu(d) M(n/d, 1, 2); d|n, d=1, 2(3) where M(n, t, s) = Sum n!/(i!j!k!); i+j+k=n, j=t(3), k=s(3).
Showing 1-6 of 6 results.

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