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A285037 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromic structures using exactly k different symbols, 1 <= k <= n/2 + 1. 14
1, 0, 1, 0, 1, 0, 2, 1, 0, 3, 1, 0, 4, 5, 1, 0, 7, 6, 1, 0, 10, 18, 7, 1, 0, 14, 25, 10, 1, 0, 21, 63, 43, 10, 1, 0, 31, 90, 65, 15, 1, 0, 42, 202, 219, 85, 13, 1, 0, 63, 301, 350, 140, 21, 1, 0, 91, 650, 1058, 618, 154, 17, 1, 0, 123, 965, 1701, 1050, 266, 28, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Permuting the symbols will not change the structure.

Equivalently, the number of n-bead aperiodic necklaces (Lyndon words) with exactly k symbols, up to permutation of the symbols, which when turned over are unchanged. When comparing with the turned over necklace a rotation is allowed but a permutation of the symbols is not.

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..2600

FORMULA

T(n, k) = Sum_{d | n} mu(n/d) * A285012(d, k).

EXAMPLE

Triangle starts:

1

0   1

0   1

0   2    1

0   3    1

0   4    5     1

0   7    6     1

0  10   18     7     1

0  14   25    10     1

0  21   63    43    10     1

0  31   90    65    15     1

0  42  202   219    85    13    1

0  63  301   350   140    21    1

0  91  650  1058   618   154   17   1

0 123  965  1701  1050   266   28   1

0 184 2016  4796  4064  1488  258  21  1

0 255 3025  7770  6951  2646  462  36  1

0 371 6220 21094 24914 12857 3222 410 26 1

0 511 9330 34105 42525 22827 5880 750 45 1

...

Example for n=6, k=2:

There are 6 inequivalent solutions to A285012(6,2) which are 001100, 010010, 000100, 001010, 001110, 010101. Of these, 010010 and 010101 have a period less than 6, so T(6,2) = 6-2 = 4.

PROG

(PARI) \\ Ach is A304972

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

T(n, k=n\2+1) = {my(A=Ach(n\2+1, k), S=matrix(n\2+1, k, n, k, stirling(n, k, 2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}

{ my(A=T(20)); for(n=1, matsize(A)[1], print(A[n, 1..n\2+1])) } \\ Andrew Howroyd, Oct 01 2019

(PARI) \\ column sequence using above code.

ColSeq(n, k=2) = { Vec(T(n, k)[, k]) } \\ Andrew Howroyd, Oct 01 2019

CROSSREFS

Columns 1..6 are: A063524, A056518, A056519, A056521, A056522, A056523.

Partial row sums include A056513, A056514, A056515, A056516, A056517.

Row sums are A285042.

Cf. A284856, A284826, A284823, A285012, A304972.

Sequence in context: A219659 A029293 A218254 * A264422 A176808 A327029

Adjacent sequences:  A285034 A285035 A285036 * A285038 A285039 A285040

KEYWORD

nonn,tabf

AUTHOR

Andrew Howroyd, Apr 08 2017

STATUS

approved

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Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)