A056459 -id:A056459 - cp's OEIS frontend

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62

Offset: 1

Andrew Howroyd, Apr 03 2017

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

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 A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)

a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

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

A056477 Number of primitive (aperiodic) palindromic structures using a maximum of three different symbols.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 3, 13, 12, 39, 36, 121, 116, 364, 351, 1088, 1080, 3280, 3237, 9841, 9800, 29510, 29403, 88573, 88440, 265716, 265356, 797121, 796796, 2391484, 2390352, 7174453, 7173360, 21523238, 21520080, 64570064, 64566684, 193710244, 193700403, 581130368, 581120880

Offset: 0

Marks R. Nester

nonn

Permuting the symbols 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]

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A007051, A124302, A056459, A284826.

a(n) = Sum_{d|n} mu(d)*A124302(ceiling(n/(2*d))) for n > 0.

a(n) = Sum_{k=1..3} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 2019

A056464 Number of primitive (aperiodic) palindromes using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 0, 6, 6, 36, 36, 150, 144, 540, 534, 1806, 1770, 5790, 5760, 18150, 17994, 55980, 55830, 170970, 170466, 519156, 518580, 1569744, 1567944, 4733670, 4732014, 14250606, 14244660, 42850116, 42844320, 128746410, 128728800

Offset: 1

Marks R. Nester

nonn

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..200

Column 3 of A327873.

Cf. A056454, A056459.

with(numtheory):with(combinat,stirling2):A056454:=n->3!*stirling2(floor((n+1)/2),3);A056464:=n->add(mobius(d)*A056454(n/d),d=divisors(n)); # C. Ronaldo

a(n) = Sum_{d|n} mu(d)*A056454(n/d).

A056494 Number of primitive (period n) periodic palindromes using a maximum of three different symbols.

Original entry on oeis.org

3, 3, 6, 12, 24, 42, 78, 144, 234, 456, 726, 1392, 2184, 4290, 6528, 12960, 19680, 39078, 59046, 117600, 177060, 353562, 531438, 1061280, 1594296, 3186456, 4782726, 9561552, 14348904, 28690752, 43046718

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

Number of aperiodic necklaces with three colors that are the same when turned over and hence have reflectional symmetry but no rotational symmetry. - Herbert Kociemba, Nov 29 2016

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 3 of A284856.

Cf. A056459.

mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)),{n,mx}]; CoefficientList[Series[gf[x,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 29 2016 *)

a(n) = Sum_{d|n} mu(d)*A038754(n/d+1).
From Herbert Kociemba, Nov 29 2016: (Start)
More generally, gf(k) is the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors.
gf(k): Sum_{n>=1} mu(n)*Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)). (End)

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A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

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A056477 Number of primitive (aperiodic) palindromic structures using a maximum of three different symbols.

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A056464 Number of primitive (aperiodic) palindromes using exactly three different symbols.

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A056494 Number of primitive (period n) periodic palindromes using a maximum of three different symbols.

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