A136184 Number of metacyclic groups of order 2^n.

1, 2, 4, 8, 12, 19, 26, 37, 48, 63, 78, 98, 117, 142, 166, 196, 225, 261, 295, 337, 377, 425, 471, 526, 578, 640, 699, 768, 834, 911, 984, 1069, 1150, 1243, 1332, 1434, 1531, 1642, 1748, 1868, 1983, 2113, 2237, 2377, 2511, 2661, 2805, 2966, 3120, 3292, 3457

Offset

1,2

Comments

For number of metacyclic groups of order p^n, prime p >= 3, see A136185.

Links

Klaus Brockhaus, Table of n, a(n) for n=1..1000 [Values computed with MAGMA]

Steven Liedahl, Enumeration of metacyclic p-groups, J. Algebra 186 (1996), no. 2, 436-446.

MAGMA Computational Algebra System, V2.14-1, Metacyclic p-groups

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

Formula

G.f.: -x*(x^10 + x^9 - x^8 + x^6 - x^3 - x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + x + 1)).

For n > 3, a(n) = (n^3 + 48*n^2 - c*n + d)/72, where c = 168 or 177 for n even/odd, and d = 432, 416 or 424 for n = 0, 1 or 2 (mod 3), according to the Liedahl paper. Since this would yield (4,4,5) for n=1,2,3, one can simply add [n<4]*(n-4) to get a formula valid for all n. - M. F. Hasler, Jan 13 2015

Example

a(3) = 4 since there are four metacyclic groups of order 2^3; they have invariants <3, 0, 0, 3, [ 8 ], >, <1, 2, 1, 1, [ 2, 4 ], >, <1, 1, 1, 2, [ 2 ], Dihedral> and <1, 1, 1, 2, [ 2 ], Quaternion> resp. (for details see MAGMA link).

Mathematica

LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1}, {1, 2, 4, 8, 12, 19, 26, 37, 48, 63, 78}, 60] (* Harvey P. Dale, May 31 2019 *)

Prog

(Magma) [ NumberOfMetacyclicPGroups(2, n): n in [1..51] ];
(PARI) A136184(n)=if(n<4, 2^(n-1), (((n+48)*n-[168, 177][1+n%2])*n+[432, 416, 424][1+n%3])/72) \\ M. F. Hasler, Jan 13 2015

Crossrefs

Cf. A136185.

Sequence in context: A171645 A125606 A192078 * A011908 A117455 A277753

Adjacent sequences: A136181 A136182 A136183 * A136185 A136186 A136187

Keyword

nonn

Author

Klaus Brockhaus, Dec 19 2007

Status

approved