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.

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

Offset: 1

Author

Andrew Howroyd, Apr 04 2017

Keywords

• nonn
• tabl

Comments

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

Examples

			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 ...
...
		

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

Crossrefs

Columns 2-6 are: A056493, A056494, A056495, A056496, A056497.

Programs

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-François Alcover, Jun 06 2017, translated from PARI *)

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(););

Formula

T(n, k) = Sum_{d | n} mu(n/d) * A284855(d, k).