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A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n. 0
1, 7, 41, 235, 1361, 7987, 47321, 281995, 1685921, 10096867, 60524201, 362972155, 2177309681, 13062280147, 78368930681, 470199300715, 2821152888641, 16926788453827, 101560343826761, 609360901747675, 3656161925798801, 21936961098633907, 131621735219132441 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Starting at a(2), consider that 2/3 - 1/2 = 1/6 with 1+6=7=a(2); 8/9 - 3/4 = 5/36 with 5+36=41=a(3); 26/27 - 7/8=19/216 with 19+216=235=a(4); 80/81 - 15/16=65/1296 with 65+1296=1361=a(5) and so forth. The numerators starting at a(3) are 5,19,65,211,665,2059,6305,... (see A001047) with 19 mod 5=4, 65 mod 19=8, 211 mod 65=16, 665 mod 211=32, 2059 mod 665=64, 6305 mod 2059=128, and so forth for higher powers of 2. - J. M. Bergot, May 09 2015

In other words, let f(n) = (3^(n-1)-1)/3^(n-1) - (2^(n-1)-1)/2^(n-1), then for n>=1 a(n) = numerator(f(n)) + denominator(f(n)). - Michel Marcus, May 29 2015

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

LINKS

Table of n, a(n) for n=1..23.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.

J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.

V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

FORMULA

G.f.: x(1-4x)/((1-2x)(1-3x)(1-6x)). a(n) = 6^(n-1)+3^(n-1)-2^(n-1).

E.g.f.: e^(6*x)+e^(3*x)-e^(2*x). [Mohammad K. Azarian, Jan 16 2009]

a(1)=1, a(2)=7, a(3)=41, a(n) = 11*a(n-1)-36*a(n-2)+36*a(n-3). [Harvey P. Dale, Nov 24 2011]

MATHEMATICA

Table[6^(n-1)+3^(n-1)-2^(n-1), {n, 25}] (* or *) LinearRecurrence[ {11, -36, 36}, {1, 7, 41}, 25] (* Harvey P. Dale, Nov 24 2011 *)

CoefficientList[Series[(1 - 4 x)/((1 - 2 x) (1 - 3 x) (1 - 6 x)), {x, 0, 33}], x] (* Vincenzo Librandi, May 12 2015 *)

PROG

(PARI) a(n)=if(n<0, 0, 6^(n-1)+3^(n-1)-2^(n-1))

(MAGMA) [6^(n-1)+3^(n-1)-2^(n-1): n in [1..30]] /* or */ I:=[1, 7, 41]; [n le 3 select I[n] else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 12 2015

CROSSREFS

Cf. A057004, A057005, A057006, A057007, A057008, A057010, A057011, A057012, A057013.

Sequence in context: A152268 A026002 A173409 * A140480 A002315 A141813

Adjacent sequences:  A057006 A057007 A057008 * A057010 A057011 A057012

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 09 2000

EXTENSIONS

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

STATUS

approved

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Last modified September 21 22:17 EDT 2019. Contains 327284 sequences. (Running on oeis4.)