

A212356


Number of terms of the cycle index polynomial Z(D_n) for the dihedral group D_n.


3



1, 2, 3, 4, 3, 5, 3, 5, 4, 5, 3, 7, 3, 5, 5, 6, 3, 7, 3, 7, 5, 5, 3, 9, 4, 5, 5, 7, 3, 9, 3, 7, 5, 5, 5, 10, 3, 5, 5, 9, 3, 9, 3, 7, 7, 5, 3, 11, 4, 7, 5, 7, 3, 9, 5, 9, 5, 5, 3, 13, 3, 5, 7, 8, 5, 9, 3, 7, 5, 9, 3, 13, 3, 5, 7, 7, 5, 9, 3, 11, 6, 5, 3, 13, 5, 5, 5, 9, 3, 13
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OFFSET

1,2


COMMENTS

See A212355 for the formula for the cycle index Z(D_n) of the dihedral group, the Harary and Palmer reference, and a link for these polynomials for n=1..15.
It seems that this is also the number of different sets of distances of n points placed on 2n equidistant points on a circle.  M. F. Hasler, Jan 28 2013


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1001


FORMULA

a(n) is the number of nonvanishing entries in row n of the array A212355.
a(1) = 1, a(2) = 2, and a(n) = tau(n) + 1, n>=3, with tau(n) the number of all divisors of n, given in A000005(n).
Except for a(1) and a(2), a(n) = A161886(n+1)  A161886(n).  Eric Desbiaux, Sep 25 2013


EXAMPLE

a(6) = 5, because tau(6) = 4. The row no. 6 of A212355 is [2,0,0,2,0,0,4,0,3,0,1] with 5 nonvanishing entries.
Illustration of a(7)=3 = number of different sets of distances of 7 points among {z=e^(i k pi/7), k=0..13}: Inequivalent configurations are, e.g.: [k]=[0,2,4,6,8,10,12] with distances {0.86777, 1.5637, 1.9499}, [k]=[0,1,2,3,4,5,6] with distances {0.44504, 0.86777, 1.2470, 1.5637, 1.8019, 1.9499}, and [k]=[0,1,2,3,4,5,7] with distances {0.44504, 0.86777, 1.2470, 1.5637, 1.8019, 1.9499, 2.0000}.  M. F. Hasler, Jan 28 2013


PROG

(PARI) A212356(n) = if(n<=2, n, 1+numdiv(n)); \\ Antti Karttunen, Sep 22 2017


CROSSREFS

Cf. A212355, A000005.
Sequence in context: A325382 A324401 A323082 * A322816 A323078 A331301
Adjacent sequences: A212353 A212354 A212355 * A212357 A212358 A212359


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Jun 02 2012


STATUS

approved



