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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. 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-Fran├žois 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.

Sequence in context: A276550 A294438 A074650 * A202064 A144955 A225624

Adjacent sequences:  A284868 A284869 A284870 * A284872 A284873 A284874

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 04 2017

STATUS

approved

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Last modified April 9 17:48 EDT 2020. Contains 333361 sequences. (Running on oeis4.)