A056362 -id:A056362

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A276543

Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

+10
15

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 13, 11, 3, 1, 0, 8, 31, 33, 16, 3, 1, 0, 14, 80, 136, 85, 27, 4, 1, 0, 21, 201, 478, 434, 171, 37, 4, 1, 0, 39, 533, 1849, 2270, 1249, 338, 54, 5, 1, 0, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1

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

OFFSET

1,8

COMMENTS

Turning over will not create a new bracelet. Permuting the colors of the beads 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 = 1..1275

FORMULA

T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).

EXAMPLE

Triangle starts:

0 1

0 1 1

0 2 2 1

0 3 5 2 1

0 5 13 11 3 1

0 8 31 33 16 3 1

0 14 80 136 85 27 4 1

0 21 201 478 434 171 37 4 1

0 39 533 1849 2270 1249 338 54 5 1

...

PROG

(PARI) \\ Ach is A304972 and R is A152175 as square matrices.

Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}

R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}

T(n)={my(M=(R(n)+Ach(n))/2); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*M[n/d, ])))}

{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

CROSSREFS

Columns 1-6 are A063524, A056366, A056367, A056368, A056369, A056370.

Partial row sums include A000046, A056362, A056363, A056364, A056365.

Row sums are A276548.

Cf. A276550, A152175, A152176, A107424, A137651, A276544, A304972.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 09 2017

STATUS

approved

A056367

Number of primitive (period n) bracelet structures using exactly three different colored beads.

+10
3

0, 0, 1, 2, 5, 13, 31, 80, 201, 533, 1401, 3822, 10395, 28859, 80201, 225286, 634265, 1796433, 5100325, 14534758, 41513402, 118879249, 341094843, 980661980, 2824223490, 8146897815, 23535345170

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

OFFSET

1,4

COMMENTS

Turning over will not create a new bracelet. Permuting the colors of the beads 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

Table of n, a(n) for n=1..27.

FORMULA

A056362(n)-A000046(n).

CROSSREFS

Column 3 of A276543.

Cf. A056304.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

A056368

Number of primitive (period n) bracelet structures using exactly four different colored beads.

+10
3

0, 0, 0, 1, 2, 11, 33, 136, 478, 1849, 6845, 26136, 98406, 373977, 1416249, 5380770, 20440250, 77794939, 296384565, 1131009781, 4321964735, 16541268223, 63400061153, 243358777620, 935431121460, 3600520732809

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

OFFSET

1,5

COMMENTS

Turning over will not create a new bracelet. Permuting the colors of the beads 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

Table of n, a(n) for n=1..26.

FORMULA

A056363(n)-A056362(n).

CROSSREFS

Column 4 of A276543.

Cf. A056305.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

A328657

Number of primitive (period n) n-bead bracelet structures which are not periodic palindromes using a maximum of three different colored beads.

+10
2

0, 0, 1, 1, 4, 9, 26, 66, 183, 488, 1342, 3690, 10220, 28470, 79720, 224230, 633040, 1793727, 5097638, 14528640, 41509364, 118868750, 341098474, 980661510, 2824295364, 8147015352, 23535794889, 68085719208, 197213728060, 571919889400, 1660412355602, 4825573629390

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

OFFSET

1,5

COMMENTS

Permuting the colors of the beads will not change the structure.

LINKS

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

FORMULA

a(n) = Sum_{k=1..3} A309784(n, k).

a(n) = A056362(n) - A056514(n).

EXAMPLE

For n <= 5, the structures are the same as in A328038.

For n = 6, the 9 bracelet structures have the patterns: AABABB, ABCCCC, ABBCCC, ABBBCB, ABBCBC, ABBCCB, ABCBBC, AABBCC, AABCBC.

CROSSREFS

Cf. A056362, A056514, A309784, A328038.

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Oct 24 2019

STATUS

approved

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