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.

User: Marks R. Nester

Marks R. Nester's wiki page.

Marks R. Nester has authored 252 sequences. Here are the ten most recent ones:

A056269 Number of primitive (aperiodic) words of length n which contain exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40800, 186480, 818280, 3498000, 14674440, 60780720, 249393480, 1016542560, 4123132800, 16664094960, 67171179600, 270232006800, 1085569963080, 4356217672800, 17466683473800
Offset: 1

Views

Author

Marks R. Nester

Keywords

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A000302, A000919, A054719.

Formula

sum mu(d)*A000919(n/d) where d|n.
Seems to be n * A056289.

A056270 Number of primitive (aperiodic) words of length n which contain exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5102880, 29607600, 165526200, 901020120, 4808987400, 25292030280, 131542740000, 678330198120, 3474970629480, 17710714165200, 89904725757000
Offset: 1

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Author

Marks R. Nester

Keywords

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A000351, A001118, A054720.

Formula

Sum mu(d)*A001118(n/d) where d|n.

A056271 Number of primitive (aperiodic) words of length n which contain exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953028720, 6711344640, 45674173440, 302899156560, 1969146930240, 12604139926560, 79694818842240, 499018753280880, 3100376788241040
Offset: 1

Views

Author

Marks R. Nester

Keywords

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A000400, A000920, A054721.

Formula

Sum mu(d)*A000920(n/d) where d|n.

A056274 Number of primitive (aperiodic) word structures of length n using a 3-ary alphabet.

Original entry on oeis.org

1, 1, 4, 12, 40, 116, 364, 1080, 3276, 9800, 29524, 88440, 265720, 796796, 2391440, 7173360, 21523360, 64566684, 193710244, 581120880, 1743391832, 5230147076, 15690529804, 47071499760, 141214768200
Offset: 1

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Author

Marks R. Nester

Keywords

Comments

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A054718.

Formula

Sum mu(d)*A007051(n/d-1) where d|n and n>0.

A056277 Number of primitive (aperiodic) word structures of length n using a 6-ary alphabet.

Original entry on oeis.org

1, 1, 4, 13, 51, 197, 875, 4096, 20643, 109246, 601491, 3402911, 19628063, 114699438, 676207572, 4010086352, 23874362199, 142508702805, 852124263683, 5101098123207, 30560194492576, 183176169456214
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A054721.

Formula

sum mu(d)*A056273(n/d) where d|n and n>0.

A056289 Number of primitive (period n) n-bead necklaces with exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 6, 48, 260, 1200, 5100, 20720, 81828, 318000, 1222870, 4675440, 17813820, 67769504, 257695800, 980240880, 3731732200, 14222737200, 54278498154, 207438936800, 793940157900, 3043140078000, 11681056021300, 44900438149248, 172824327151140, 666070256468960
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over the necklace is not allowed.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Crossrefs

Cf. A027377.
Column k=4 of A254040.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
a:= n-> add(b(n, 4-j)*binomial(4, j)*(-1)^j, j=0..4):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2015
  • Mathematica
    b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n]; a[n_] := Sum[b[n, 4-j]*Binomial[4, j]*(-1)^j, {j, 0, 4}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 20 2017, after Alois P. Heinz *)

Formula

a(n) = Sum_{d|n} mu(d)*A056284(n/d).

A056302 Number of primitive (period n) n-bead necklace structures using a maximum of six different colored beads.

Original entry on oeis.org

1, 1, 2, 5, 11, 39, 125, 532, 2301, 11010, 54681, 284023, 1509851, 8194902, 45080652, 250641356, 1404374247, 7917209005, 44848645457, 255055220735, 1455247360000, 8326191235902, 47752990403133
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.
a(n) = A107424(n, 1)+A107424(n, 2)+...+A107424(n, 6). - David Wasserman, May 31 2005

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A032164.

Formula

sum mu(d)*A056294(n/d) where d|n and n>0.

A056303 Number of primitive (period n) n-bead necklace structures using exactly two different colored beads.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1091, 2048, 3855, 7280, 13797, 26214, 49929, 95325, 182361, 349520, 671088, 1290555, 2485504, 4793490, 9256395, 17895679, 34636833, 67108864, 130150493, 252645135, 490853403, 954437120, 1857283155
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.
Identical to A000048 for n>1.
Number of binary Lyndon words of length n with an odd number of zeros. - Joerg Arndt, Oct 26 2015

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Column 2 of A107424.
Cf. A000007, A000048, A001037, A056295.

Programs

  • PARI
    vector(100, n, sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n)-!(n-1)) \\ Altug Alkan, Oct 26 2015
  • Python
    from sympy import divisors, mobius
def a000048(n): return 1 if n<1 else sum([mobius(d)*2**(n/d) for d in divisors(n) if d%2 == 1])/(2*n)
def a(n): return a000048(n) - 0**(n - 1) # Indranil Ghosh, Apr 28 2017

Formula

a(n) = Sum mu(d)*A056295(n/d) where d divides n.
a(n) = A000048(n) - A000007(n-1).

A056305 Number of primitive (period n) n-bead necklace structures using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 1, 2, 13, 50, 220, 866, 3435, 13250, 51061, 194810, 742601, 2823764, 10738660, 40843370, 155493872, 592614050, 2261622287, 8643289484, 33080907357, 126797503250, 486710920300, 1870851589552
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Column 4 of A107424.
Cf. A056289.

Formula

Sum mu(d)*A056297(n/d) where d|n. Alternatively, A056300(n)-A002075(n)..

A056306 Number of primitive (period n) n-bead necklace structures using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 20, 136, 773, 4280, 22430, 115097, 577577, 2863207, 14051163, 68515378, 332514803, 1608799915, 7767857090, 37460384315, 180536313527, 869901375049, 4192038616700, 20208367780744
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Column 5 of A107424.
Cf. A056290.

Formula

Sum mu(d)*A056298(n/d) where d|n. Alternatively, A056301(n)-A056300(n)..

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