A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.

1, 1, 1, 3, 1, 3, 2, 1, 5, 0, 2, 1, 5, 0, 0, 4, 1, 7, 2, 0, 0, 2, 1, 7, 0, 0, 0, 0, 6, 1, 9, 0, 2, 0, 0, 0, 4, 1, 9, 2, 0, 0, 0, 0, 0, 6, 1, 11, 0, 0, 4, 0, 0, 0, 0, 4, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 13, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 15, 0, 0, 0, 0, 6, 0, 0

Offset: 1

Author

Roger Cuculière, Oct 29 2000

Keywords

• nonn
• tabf
• easy
• nice

Comments

Note that D_2 equals the cyclic group of order 2.

Examples

			1,1;
1,3;
1,3,2;
1,5,0,2;
1,5,0,0,4; ...
		

Crossrefs

Cf. A057731, A054522, A057740.

Programs

Mathematica

t[n_, k_] /; k != 2 && ! Divisible[n, k] = 0; t[n_, k_] /; k != 2 && Divisible[n, k] := EulerPhi[k]; t[n_, 2] := n + 1 - Mod[n, 2]; Flatten[Table[t[n, k], {n, 1, 14}, {k, 1, If[n == 1, 2, n]}]] (* Jean-François Alcover, Jun 19 2012, from formula *)
row[n_] := (orders = PermutationOrder /@ GroupElements[DihedralGroup[n]]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

Formula

If k<>2 and k does not divide n, this number is 0; if k<>2 and k divides n, this number is phi(k), where phi is the Euler totient function; if k=2, this number is n for odd n and n+1 for even n.

Extensions

More terms from James Sellers, Oct 30 2000