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A284856 Array read by antidiagonals: T(n,k) = number of aperiodic necklaces (Lyndon words) with n beads and k colors that are the same when turned over. 8
1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 6, 3, 0, 6, 10, 12, 12, 6, 0, 7, 15, 20, 30, 24, 7, 0, 8, 21, 30, 60, 60, 42, 14, 0, 9, 28, 42, 105, 120, 138, 78, 18, 0, 10, 36, 56, 168, 210, 340, 252, 144, 28, 0, 11, 45, 72, 252, 336, 705, 620, 600, 234, 39, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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) * A284855(d, k).

EXAMPLE

Table starts:

1  2   3    4    5     6     7      8      9     10 ...

0  1   3    6   10    15    21     28     36     45 ...

0  2   6   12   20    30    42     56     72     90 ...

0  3  12   30   60   105   168    252    360    495 ...

0  6  24   60  120   210   336    504    720    990 ...

0  7  42  138  340   705  1302   2212   3528   5355 ...

0 14  78  252  620  1290  2394   4088   6552   9990 ...

0 18 144  600 1800  4410  9408  18144  32400  54450 ...

0 28 234 1008 3100  7740 16758  32704  58968  99900 ...

0 39 456 2490 9240 26985 66864 146916 294480 548955 ...

...

MATHEMATICA

b[d_, k_] := If[EvenQ[d], (k^(d/2) + k^(d/2 + 1))/2, k^((d + 1)/2)];

a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k] &];

Table[a[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Jun 06 2017, translated from PARI *)

PROG

(PARI)

b(d, k) = if(d % 2 == 0, (k^(d/2) + k^(d/2+1))/2, k^((d+1)/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: A056493, A056494, A056495, A056496, A056497.

Sequence in context: A321980 A208544 A208535 * A276550 A294438 A074650

Adjacent sequences:  A284853 A284854 A284855 * A284857 A284858 A284859

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 04 2017

STATUS

approved

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)