A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

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

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.

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