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
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Dec 19 2007
STATUS
approved