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A136185 Number of metacyclic groups of order p^n, prime p >= 3. 3
1, 2, 3, 5, 7, 11, 14, 20, 25, 33, 40, 51, 60, 74, 86, 103, 118, 139, 157, 182, 204, 233, 259, 293, 323, 362, 397, 441, 481, 531, 576, 632, 683, 745, 802, 871, 934, 1010, 1080, 1163, 1240, 1331, 1415, 1514, 1606, 1713, 1813, 1929, 2037, 2162, 2279, 2413, 2539 (list; graph; refs; listen; history; text; internal format)
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

Cf. A136184.

Sequence in context: A036608 A309097 A309098 * A319471 A218506 A238659

Adjacent sequences:  A136182 A136183 A136184 * A136186 A136187 A136188

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Dec 19 2007

STATUS

approved

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Last modified September 24 22:16 EDT 2020. Contains 337325 sequences. (Running on oeis4.)