A056314 -id:A056314 - cp's OEIS frontend

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

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

Andrew Howroyd, Apr 04 2017

A string and its reverse are considered to be equivalent.

			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 ...
...

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]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A045625, A056314, A056315, A056316, A056317.

Cf. A277504, A143324.

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 *)

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(););

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

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

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

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]

Column 4 of A284871.

Cf. A045625, A056314.

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

A056332 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of three different colors.

Original entry on oeis.org

1, 1, 3, 8, 24, 65, 195, 564, 1677, 4976, 14883, 44452, 133224, 399113, 1196808, 3588840, 10764960, 32289855, 96864963, 290580040, 871725426, 2615132465, 7845353475, 23535926760, 70607649816, 211822550576

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

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]

Cf. A001998, A056314.

a1998[n_] := If[OddQ[n], (1/4)*(3^n + 4*3^((n-1)/2) + 1), (1/4)*(3^n + 2*3^(n/2) + 1)];
a[n_] := DivisorSum[n, MoebiusMu[#] a1998[n/#-1]&];
Array[a, 26] (* Jean-François Alcover, Jun 29 2018 *)

a(n) = Sum mu(d)*A001998(n/d-1) where d|n.

A056319 Number of primitive (aperiodic) reversible strings with n beads using exactly three different colors.

Original entry on oeis.org

0, 0, 3, 18, 78, 270, 921, 2898, 9147, 27987, 85773, 259557, 785778, 2366892, 7128120, 21425040, 64382550, 193316685, 580372293, 1741819245, 5227115454, 15684238080, 47059266081, 141189599250

Offset: 1

Marks R. Nester

nonn

Each string and its reverse are considered to be equivalent.

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]

Cf. A056314.

Sum mu(d)*A056310(n/d) where d|n.

cp's OEIS Frontend

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

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

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A056332 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of three different colors.

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A056319 Number of primitive (aperiodic) reversible strings with n beads using exactly three different colors.

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