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 1-10 of 10 results.

A056391 Number of step shifted (decimated) sequence structures using a maximum of two different symbols.

Original entry on oeis.org

1, 2, 3, 6, 6, 20, 14, 48, 52, 140, 108, 624, 352, 1400, 2172, 4464, 4116, 22112, 14602, 68016, 88376, 209936, 190746, 1075200, 839128, 2797000, 3730584, 11276704, 9587580, 67195520, 35792568
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the structure.
Also, number of circulant digraphs on n vertices up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic (see A049297). - Andrew Howroyd, Apr 20 2017

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.

Links

Crossrefs

Row 2 of A285522.
Cf. A056371, A049297, A285620, A049287, A288620.

Programs

  • Mathematica
    a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; a[n_] := a[2, n]/2; Array[a, 40] (* Jean-François Alcover, Jun 12 2017 *)
  • PARI
    a(n)=sum(k=1, n, if(gcd(k, n)==1, 2^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n); \\ Andrew Howroyd, Apr 20 2017
  • PARI
    \\ alternative using Polya enumeration functions (see attachment)
a(n) = NonequivalentStructs(StepShiftPerms(n),2); \\ Andrew Howroyd, Oct 01 2017

Formula

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
a(n) = A056371(n) / 2. - Andrew Howroyd, Apr 20 2017
a(n) = A288620(n, 2) + 1. - Andrew Howroyd, Jun 13 2017

A288627 Triangle read by rows: T(n,k) = number of step cyclic shifted sequence structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 3, 1, 1, 1, 7, 14, 11, 3, 1, 1, 4, 11, 13, 6, 1, 1, 1, 13, 52, 83, 52, 18, 3, 1, 1, 10, 72, 162, 148, 59, 13, 2, 1, 1, 25, 274, 930, 1140, 630, 171, 28, 3, 1, 1, 14, 281, 1369, 2306, 1681, 612, 118, 14, 1, 1
Offset: 1

Views

Author

Andrew Howroyd, Jun 11 2017

Keywords

Comments

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent. Permuting the symbols will not change the structure.

Examples

			Triangle begins
1;
1,  1;
1,  1,   1;
1,  3,   2,   1;
1,  2,   3,   1,    1;
1,  7,  14,  11,    3,   1;
1,  4,  11,  13,    6,   1,   1;
1, 13,  52,  83,   52,  18,   3,  1;
1, 10,  72, 162,  148,  59,  13,  2, 1;
1, 25, 274, 930, 1140, 630, 171, 28, 3, 1;
...

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

Columns 2-6 are A056434, A056435, A056436, A056437, A056438.
Row sums are A288628.
Partial row sums include A056429, A056430, A056431, A056432, A056433.
Cf. A056391, A056371, A288620, A285522, A285548, A132191.

Programs

  • PARI
    \\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(CyclicStepShiftPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

A056392 Number of step shifted (decimated) sequence structures using a maximum of three different symbols.

Original entry on oeis.org

1, 2, 4, 10, 14, 70, 68, 332, 577, 2510, 2980, 23372, 22218, 133150, 300964, 909382, 1345634, 10767202, 10762820, 72957100, 145362932, 523029526, 713213956, 5893709440, 7060765733, 35303782550
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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. A056372, A288620.

Formula

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
a(n) = Sum_{k=1..3} A288620(n, k). - Andrew Howroyd, Jun 13 2017

A056393 Number of step shifted (decimated) sequence structures using a maximum of four different symbols.

Original entry on oeis.org

1, 2, 4, 11, 17, 107, 131, 811, 1893, 11107, 17599, 179371, 233449, 1866057, 5603787, 22469291, 44744047, 477262537, 636308685, 5732457131, 15272176697, 73301054891, 133274359129, 1466413263531
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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. A056373, A288620.

Formula

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
a(n) = Sum_{k=1..4} A288620(n, k). - Andrew Howroyd, Jun 13 2017

A288621 Number of step shifted (decimated) sequence structures of length n using an infinite alphabet.

Original entry on oeis.org

1, 2, 4, 11, 18, 117, 162, 1205, 3621, 29271, 68086, 1069614, 2305158, 31825670, 172974370, 1311400891, 5179123522, 113680235941, 324041807958, 6466011855715, 39572612670818, 450671611457619, 2006909406239780, 55745145216877102, 231929517217884081
Offset: 1

Views

Author

Andrew Howroyd, Jun 11 2017

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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

Row sums of A288620.
Cf. A056391, A056371, A288628.

Programs

  • PARI
    \\ see A056391 for Polya enumeration functions
a(n) = NonequivalentStructs(StepShiftPerms(n), n); \\ Andrew Howroyd, Oct 14 2017

A056396 Number of step shifted (decimated) sequence structures using exactly two different symbols.

Original entry on oeis.org

0, 1, 2, 5, 5, 19, 13, 47, 51, 139, 107, 623, 351, 1399, 2171, 4463, 4115, 22111, 14601, 68015, 88375, 209935, 190745, 1075199, 839127, 2796999, 3730583, 11276703, 9587579, 67195519, 35792567
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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 2 of A288620.
Cf. A056376.

Formula

A056391(n)-1.

A056397 Number of step shifted (decimated) sequence structures using exactly three different symbols.

Original entry on oeis.org

0, 0, 1, 4, 8, 50, 54, 284, 525, 2370, 2872, 22748, 21866, 131750, 298792, 904918, 1341518, 10745090, 10748218, 72889084, 145274556, 522819590, 713023210, 5892634240, 7059926605, 35300985550, 70604008204
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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 3 of A288620.
Cf. A056377.

Formula

A056392(n)-A056391(n).

A056398 Number of step shifted (decimated) sequence structures using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 1, 3, 37, 63, 479, 1316, 8597, 14619, 155999, 211231, 1732907, 5302823, 21559909, 43398413, 466495335, 625545865, 5659500031, 15126813765, 72778025365, 132561145173, 1460519554091, 2338565021382
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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 A288620.
Cf. A056378.

Formula

A056393(n)-A056392(n).

A056399 Number of step shifted (decimated) sequence structures using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 1, 9, 26, 299, 1183, 10701, 24736, 349241, 626053, 6680979, 26364610, 137247254, 353304459, 4826418027, 8199440662, 93679529503, 315943737331, 1913783074461, 4382586624256, 60629067776483
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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 A288620.
Cf. A056379.

Formula

A056394(n)-A056393(n).

A056400 Number of step shifted (decimated) sequence structures using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 83, 454, 5761, 17998, 334797, 777107, 10574995, 52611728, 342199022, 1094122103, 18448004619, 38504610244, 538330833767, 2215487105217, 16330536710131, 45407723468330
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the 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 6 of A288620.
Cf. A056380.

Formula

A056395(n)-A056394(n).
Showing 1-10 of 10 results.

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