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

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, 39, 14, 28, 15, 62, 16, 32, 17, 55, 18, 36, 19, 74, 20, 40, 21, 63, 22, 44, 23, 106, 24, 48, 25, 75, 26, 52, 27, 106, 28, 56, 29, 103, 30, 60, 31, 142, 32, 64, 33, 99

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 20 2001

Keywords

• nonn

Examples

			a(3) = 1 because the group is isomorphic to S_3 and the table is : 1, 1, 1 1,-1, 1 2, 0,-1
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A060762.

Programs

Mathematica

a[n_] := Count[FiniteGroupData[{"DihedralGroup", n}, "CharacterTable"], 0, 2]; Array[a, 100] (* Jean-François Alcover, Oct 08 2016 *)

Sage

def A061721(n) :
    if n % 2 == 1 : return (n - 1) // 2
    if n % 4 == 2 : return n - 2
    numzeros = n - 2
    np = n // 4
    for m in range(1, n // 2) :
        t = lcm(m, np)
        if (t // np) % 2 == 1 :
            maxmul = m * n // 2
            numzeros += (maxmul // t) - (maxmul // (2*t))
    return numzeros
# Eric M. Schmidt, Jul 04 2012

Formula

For odd n, a(n) = (n-1)/2.

For n = 2 (mod 4), a(n) = n - 2. - Eric M. Schmidt, Jul 04 2012

Extensions

More terms from Eric M. Schmidt, Jul 04 2012