A212356 Number of terms of the cycle index polynomial Z(D_n) for the dihedral group D_n.

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

Offset: 1

Author

Wolfdieter Lang, Jun 02 2012

Keywords

• nonn
• easy

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

Examples

			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 non-vanishing 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
		

Links

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

Crossrefs

Cf. A212355, A000005.

Programs

PARI

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

Formula

a(n) is the number of non-vanishing 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