OFFSET
1,2
COMMENTS
For number of metacyclic groups of order 2^n see A136184.
LINKS
Klaus Brockhaus, Table of n, a(n) for n=1..1000 [Values computed with MAGMA]
Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
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^7 - 2*x^5 + x^3 + x^2 - x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + x + 1)).
a(n) = (n^3 + 12*n^2 + c*n + d)/72, where c = 12 or 3 for n even/odd, and d = 72, 56 or 64 for n = 0, 1 or 2 (mod 3), according to the Liedahl paper. - M. F. Hasler, Jan 13 2015
EXAMPLE
a(4) = 5 since there are five metacyclic groups of order p^4; they have invariants <4, 0, 0, 4, [ p^4 ], >, <1, 2, 1, 2, [], >, <1, 2, 2, 2, [], >, <2, 2, 2, 2, [], > and <1, 3, 1, 1, [], > resp. (for details see MAGMA link).
MATHEMATICA
Rest@ CoefficientList[Series[-x (x^7 - 2 x^5 + x^3 + x^2 - x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + x + 1)), {x, 0, 53}], x] (* Michael De Vlieger, Nov 04 2019 *)
PROG
(Magma) [ NumberOfMetacyclicPGroups(3, n): n in [1..53] ];
(PARI) A136185(n)=(((n+12)*n+[12, 3][1+n%2])*n+[72, 56, 64][1+n%3])/72 \\ M. F. Hasler, Jan 13 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Dec 19 2007
STATUS
approved