cp's OEIS Frontend

A061721 Number of zeros in the character table of the dihedral group with 2n elements.

%I A061721 #19 Mar 07 2020 05:38:04
%S A061721 0,0,1,3,2,4,3,10,4,8,5,15,6,12,7,26,8,16,9,27,10,20,11,42,12,24,13,
%T A061721 39,14,28,15,62,16,32,17,55,18,36,19,74,20,40,21,63,22,44,23,106,24,
%U A061721 48,25,75,26,52,27,106,28,56,29,103,30,60,31,142,32,64,33,99
%N A061721 Number of zeros in the character table of the dihedral group with 2n elements.
%H A061721 Eric M. Schmidt, <a href="/A061721/b061721.txt">Table of n, a(n) for n = 1..10000</a>
%F A061721 For odd n, a(n) = (n-1)/2.
%F A061721 For n = 2 (mod 4), a(n) = n - 2. - _Eric M. Schmidt_, Jul 04 2012
%e A061721 a(3) = 1 because the group is isomorphic to S_3 and the table is : 1, 1, 1 1,-1, 1 2, 0,-1
%t A061721 a[n_] := Count[FiniteGroupData[{"DihedralGroup", n}, "CharacterTable"], 0, 2]; Array[a, 100] (* _Jean-François Alcover_, Oct 08 2016 *)
%o A061721 (Sage)
%o A061721 def A061721(n) :
%o A061721     if n % 2 == 1 : return (n - 1) // 2
%o A061721     if n % 4 == 2 : return n - 2
%o A061721     numzeros = n - 2
%o A061721     np = n // 4
%o A061721     for m in range(1, n // 2) :
%o A061721         t = lcm(m, np)
%o A061721         if (t // np) % 2 == 1 :
%o A061721             maxmul = m * n // 2
%o A061721             numzeros += (maxmul // t) - (maxmul // (2*t))
%o A061721     return numzeros
%o A061721 # _Eric M. Schmidt_, Jul 04 2012
%Y A061721 Cf. A060762.
%K A061721 nonn
%O A061721 1,4
%A A061721 Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 20 2001
%E A061721 More terms from _Eric M. Schmidt_, Jul 04 2012