A056391 Number of step shifted (decimated) sequence structures using a maximum of two different symbols.

1, 2, 3, 6, 6, 20, 14, 48, 52, 140, 108, 624, 352, 1400, 2172, 4464, 4116, 22112, 14602, 68016, 88376, 209936, 190746, 1075200, 839128, 2797000, 3730584, 11276704, 9587580, 67195520, 35792568

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

See A056371 for an explanation of step shifts. Permuting the symbols will not change the structure.

Also, number of circulant digraphs on n vertices up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic (see A049297). - Andrew Howroyd, Apr 20 2017

References

• M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Andrew Howroyd)
• Andrew Howroyd, Polya Enumeration in PARI (many sequences included)
• Marks R. Nester, Mathematical investigations of some plant interaction designs, Chapter 2, Finite and Periodic Sequences, plus Notes and Errata.

Crossrefs

Row 2 of A285522.

Cf. A056371, A049297, A285620, A049287, A288620.

Programs

Mathematica

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; a[n_] := a[2, n]/2; Array[a, 40] (* Jean-François Alcover, Jun 12 2017 *)

PARI

a(n)=sum(k=1, n, if(gcd(k, n)==1, 2^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n); \\ Andrew Howroyd, Apr 20 2017

PARI

\\ alternative using Polya enumeration functions (see attachment)
a(n) = NonequivalentStructs(StepShiftPerms(n),2); \\ Andrew Howroyd, Oct 01 2017

Formula

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = A056371(n) / 2. - Andrew Howroyd, Apr 20 2017

a(n) = A288620(n, 2) + 1. - Andrew Howroyd, Jun 13 2017