A056332 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of three different colors.

1, 1, 3, 8, 24, 65, 195, 564, 1677, 4976, 14883, 44452, 133224, 399113, 1196808, 3588840, 10764960, 32289855, 96864963, 290580040, 871725426, 2615132465, 7845353475, 23535926760, 70607649816, 211822550576

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

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]

Crossrefs

Cf. A001998, A056314.

Programs

Mathematica

a1998[n_] := If[OddQ[n], (1/4)*(3^n + 4*3^((n-1)/2) + 1), (1/4)*(3^n + 2*3^(n/2) + 1)];
a[n_] := DivisorSum[n, MoebiusMu[#] a1998[n/#-1]&];
Array[a, 26] (* Jean-François Alcover, Jun 29 2018 *)

Formula

a(n) = Sum mu(d)*A001998(n/d-1) where d|n.