A056462 -id:A056462

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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).

+10
10

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

FORMULA

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

EXAMPLE

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 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).

MATHEMATICA

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 *)

PROG

(PARI)

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(); )

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 03 2017

STATUS

approved

A056467

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

+10
4

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 720, 720, 15120, 15120, 191520, 191520, 1905120, 1905120, 16435440, 16435440, 129230640, 129229920, 953029440, 953028720, 6711344640, 6711329520, 45674188560, 45674173440, 302899156560, 302898965040, 1969147121760, 1969146930240

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,11

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

Andrew Howroyd, Table of n, a(n) for n = 1..200

FORMULA

a(n) = Sum_{d|n} mu(d)*A056457(n/d).

CROSSREFS

Column 6 of A327873.

Cf. A056457, A056462.

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

Terms a(28) and beyond from Andrew Howroyd, Sep 29 2019

STATUS

approved

A056480

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

+10
4

1, 1, 0, 1, 1, 4, 3, 14, 13, 50, 47, 202, 197, 875, 861, 4105, 4096, 20647, 20593, 109298, 109246, 601476, 601289, 3403126, 3402911, 19628059, 19627188, 114700263, 114699438, 676207627, 676203467, 4010090462, 4010086352, 23874361996, 23874341552, 142508723632

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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]

LINKS

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

FORMULA

a(n) = Sum_{d|n} mu(d)*A056471(n/d) for n > 0.

a(n) = Sum_{k=1..6} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

CROSSREFS

Cf. A056462, A056471, A284826.

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

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

STATUS

approved

A056497

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

+10
1

6, 15, 30, 105, 210, 705, 1290, 4410, 7740, 26985, 46650, 162435, 279930, 978465, 1679370, 5874120, 10077690, 35263440, 60466170, 211604295, 362795730, 1269743025, 2176782330, 7618570470, 13060693800

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

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

Table of n, a(n) for n=1..25.

FORMULA

a(n) = Sum_{d|n} mu(d)*A056488(n/d).

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)

EXAMPLE

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

MATHEMATICA

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, 6], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *)

CROSSREFS

Column 6 of A284856.

Cf. A056462.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

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