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.

A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0
Offset: 1

Views

Author

Andrew Howroyd, Apr 04 2017

Keywords

Comments

A string and its reverse are considered to be equivalent.

Examples

			Table starts:
1   2    3     4      5      6       7       8 ...
0   1    3     6     10     15      21      28 ...
0   4   15    36     70    120     189     280 ...
0   7   39   126    310    645    1197    2044 ...
0  18  132   540   1620   3990    8568   16632 ...
0  29  357  2034   7790  23295   58779  131012 ...
0  70 1131  8316  39370 140610  412965 1050616 ...
0 126 3276 32760 195300 839790 2882376 8388576 ...
...

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 A045625, A056314, A056315, A056316, A056317.
Cf. A277504, A143324.

Programs

  • Mathematica
    b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)
  • PARI
    b(n,k) = (k^n + k^(ceil(n/2))) / 2;
a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););

Formula

T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.

A056314 Number of primitive (aperiodic) reversible strings with n beads using a maximum of three different colors.

Original entry on oeis.org

3, 3, 15, 39, 132, 357, 1131, 3276, 9945, 29508, 88935, 265668, 798252, 2391441, 7177584, 21523320, 64579920, 193709763, 581160255, 1743392040, 5230264026, 15690529437, 47071855131, 141214764600
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent.

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 A284871.
Cf. A045625.

Formula

Sum mu(d)*A032120(n/d) where d|n.

A056315 Number of primitive (aperiodic) reversible strings with n beads using a maximum of four different colors.

Original entry on oeis.org

4, 6, 36, 126, 540, 2034, 8316, 32760, 131544, 524250, 2099196, 8388450, 33562620, 134217594, 536903100, 2147483520, 8590065660, 34359735816, 137439477756, 549755813250, 2199025344348, 8796093020154
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent.

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 A284871.
Cf. A045625, A056314.

Formula

Sum mu(d)*A032121(n/d) where d|n.

A056316 Number of primitive (aperiodic) reversible strings with n beads using a maximum of five different colors.

Original entry on oeis.org

5, 10, 70, 310, 1620, 7790, 39370, 195300, 978050, 4882740, 24421870, 122069940, 610390620, 3051757490, 15258982680, 76293945000, 381470703120, 1907348623450, 9536748046870, 47683715818440
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent.

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 A284871.
Cf. A045625.

Formula

Sum mu(d)*A032122(n/d) where d|n.

A056317 Number of primitive (aperiodic) reversible strings with n beads using a maximum of six different colors.

Original entry on oeis.org

6, 15, 120, 645, 3990, 23295, 140610, 839790, 5042610, 30232965, 181421850, 1088390415, 6530486970, 39182081385, 235093327980, 1410554953080, 8463334761210, 50779978307010, 304679900238330
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent.

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 A284871.
Cf. A045625.

Formula

sum mu(d)*A056308(n/d) where d|n.

A056331 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of two different colors.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 35, 66, 133, 261, 527, 1032, 2079, 4123, 8244, 16440, 32895, 65639, 131327, 262380, 524762, 1049071, 2098175, 4195230, 8390646, 16779231, 33558392, 67112892, 134225919, 268443306
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent. Permuting the colors 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. A045625.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] (2^(n/#-2) + 2^(Floor[n/#/2]-1))&];
Array[a, 30] (* Jean-François Alcover, Jun 29 2018 *)

Formula

a(n) = sum mu(d)*A005418(n/d) where d divides n.

A056318 Number of primitive (aperiodic) reversible strings with n beads using exactly two different colors.

Original entry on oeis.org

0, 1, 4, 7, 18, 29, 70, 126, 266, 507, 1054, 2037, 4158, 8183, 16488, 32760, 65790, 131026, 262654, 524265, 1049524, 2097119, 4196350, 8388450, 16781292, 33554367, 67116784, 134217657, 268451838
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent. Identical to A045625 for n>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]

Crossrefs

Cf. A045625.

Formula

Sum mu(d)*A056309(n/d) where d|n.
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.