A056314 -id:A056314

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A284871

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

+10
8

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

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

OFFSET

1,2

COMMENTS

A string and its reverse are considered to be equivalent.

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^n + k^(ceiling(n/2))) / 2.

EXAMPLE

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 ...

...

MATHEMATICA

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 *)

PROG

(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(); );

CROSSREFS

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

Cf. A277504, A143324.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 04 2017

STATUS

approved

A056315

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

+10
3

4, 6, 36, 126, 540, 2034, 8316, 32760, 131544, 524250, 2099196, 8388450, 33562620, 134217594, 536903100, 2147483520, 8590065660, 34359735816, 137439477756, 549755813250, 2199025344348, 8796093020154

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

OFFSET

1,1

COMMENTS

A string and its reverse are considered to be equivalent.

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..22.

FORMULA

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

CROSSREFS

Column 4 of A284871.

Cf. A045625, A056314.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

A056332

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

+10
3

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

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

OFFSET

1,3

COMMENTS

A string and its reverse are considered to be equivalent. Permuting the colors 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

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A001998, A056314.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

A056319

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

+10
1

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

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

OFFSET

1,3

COMMENTS

Each string and its reverse are considered to be equivalent.

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..24.

FORMULA

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

CROSSREFS

Cf. A056314.

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

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