OFFSET
1,8
COMMENTS
Counts contiguously substituted cycloalkane polyols (CSCPs).
From Ed Wynn and Andrew Howroyd, May 22 2021: (Start)
Consider a bracelet of n beads, each colored blue on one side and red on the other. Turning the bracelet over has the effect of simultaneously swapping the colors and reversing the order of the beads. For example, rrbbrb when turned over becomes rbrrbb. The total number of such bracelets is counted by A053656(n).
Swapping the colors is equivalent to reversing the order of the beads. For example, rrbbrb becomes bbrrbr which when turned over is brbbrr. A bracelet may or may not be the same as its reversal (or complement). The case of equality is counted by A256217(n) and the remainder can be divided into "chiral" pairs which are the reverse of each other and counted by this sequence. a(n) is then the number of pairs of two-colored n-bead bracelets that are equal under reversal but unequal up to rotation and turning over.
In chemical terms, these pairs are called "enantiomeric pairs". The example of rrbbrb corresponds to a pair of "chiral" chemical molecules: L-chiro-inositol and R-chiro-inositol.
a(n) is also half the number of nonisomorphic orientations of the n-cycle graph which are not self-converse. Again the self-converse orientations are counted by A256217(n) and the total by A053656(n).
(End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..3335
D. Bundala, M. Codish, L. Cruz-Filipe et al., Optimal-Depth Sorting Networks, arXiv preprint arXiv:1412.5302 [cs.DS], 2014. See Table 4 and associated comments.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366 | doi:10.1246/bcsj.20160369. See Table 8.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Table 2 (E_c).
FORMULA
EXAMPLE
From Ed Wynn and Andrew Howroyd, May 22 2021: (Start)
The a(6) = 1 pair of bracelets are rrbbrb and its complement bbrrbr. These two are not the same under simultaneous reversal and swapping the colors (rrbbrb is equivalent to rbrrbb which is not the same as bbrrbr by rotation).
Replacing r with ->- and b with -<- gives two distinct orientations of the cycle:
->-.->-.-<-.-<-.->-.-<- : ->-.-<-.->-.->-.-<-.-<-
| | : | |
-----------.----------- : -----------.-----------
These two might be written in shorthand as >><<>< and <<>><>.
The a(8) = 4 pairs of bracelets are rrrrbrbb, rrrbrrbb, rrrbrbbb, rrbrbrbb and their complements.
(End)
MATHEMATICA
Table[Total[EulerPhi[#] 2^(n/#) & /@ Divisors[n]]/(2 n) + 2^(n/2 - 2) (1 - Mod[n, 2]) - If[n < 1, Boole[n == 0], 2^Quotient[n, 2]/2 + DivisorSum[n, EulerPhi[2 #] 2^(n/#) &]/(4 n)], {n, 35}] (* Michael De Vlieger, Sep 05 2015, after Jean-François Alcover at A053656 and Michael Somos at A000011 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 26 2015
STATUS
approved