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A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n. 0

%I #31 Sep 08 2022 08:45:01

%S 1,7,41,235,1361,7987,47321,281995,1685921,10096867,60524201,

%T 362972155,2177309681,13062280147,78368930681,470199300715,

%U 2821152888641,16926788453827,101560343826761,609360901747675,3656161925798801,21936961098633907,131621735219132441

%N Number of conjugacy classes of subgroups of index 3 in free group of rank n.

%C 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

%C 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

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

%H J. H. Kwak and J. Lee, <a href="http://dx.doi.org/10.1002/(SICI)1097-0118(199610)23:2%3C105::AID-JGT1%3E3.0.CO;2-X">Enumeration of connected graph coverings</a>, J. Graph Th., 23 (1996), 105-109.

%H J. H. Kwak and J. Lee, <a href="http://com2mac.postech.ac.kr/resorce/Lecture_text.htm">Enumeration of graph coverings and surface branched coverings</a>, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.

%H V. A. Liskovets, <a href="http://dx.doi.org/10.1023/A:1005950823566">Reductive enumeration under mutually orthogonal group actions</a>, Acta Applic. Math., 52 (1998), 91-120.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-36,36).

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

%F E.g.f.: e^(6*x)+e^(3*x)-e^(2*x). [_Mohammad K. Azarian_, Jan 16 2009]

%F 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]

%t 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 *)

%t CoefficientList[Series[(1 - 4 x)/((1 - 2 x) (1 - 3 x) (1 - 6 x)), {x, 0, 33}], x] (* _Vincenzo Librandi_, May 12 2015 *)

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

%o (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

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Sep 09 2000

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

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