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

%I A212356 #22 Mar 17 2018 04:03:15
%S A212356 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,
%T A212356 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,
%U A212356 7,5,9,3,13,3,5,7,7,5,9,3,11,6,5,3,13,5,5,5,9,3,13
%N A212356 Number of terms of the cycle index polynomial Z(D_n) for the dihedral group D_n.
%C A212356 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.
%C A212356 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
%H A212356 Antti Karttunen, <a href="/A212356/b212356.txt">Table of n, a(n) for n = 1..1001</a>
%F A212356 a(n) is the number of non-vanishing entries in row n of the array A212355.
%F A212356 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).
%F A212356 Except for a(1) and a(2), a(n) = A161886(n+1) - A161886(n). - _Eric Desbiaux_, Sep 25 2013
%e A212356 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.
%e A212356 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
%o A212356 (PARI) A212356(n) = if(n<=2,n,1+numdiv(n)); \\ _Antti Karttunen_, Sep 22 2017
%Y A212356 Cf. A212355, A000005.
%K A212356 nonn,easy
%O A212356 1,2
%A A212356 _Wolfdieter Lang_, Jun 02 2012