A049294 Number of subgroups of index 3 in free group of rank n+1.

1, 13, 97, 625, 3841, 23233, 139777, 839425, 5038081, 30231553, 181395457, 1088385025, 6530334721, 39182057473, 235092443137, 1410554855425, 8463329525761, 50779977940993, 304679869218817, 1828079218458625

Offset: 0

Author

Valery A. Liskovets

Keywords

• easy
• nice
• nonn

References

• P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
• V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
• Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Crossrefs

Cf. A003319, A027837, A049290, A049291, A049292, A049293, A049295.

Programs

Mathematica

LinearRecurrence[{9,-20,12},{1,13,97},20] (* Harvey P. Dale, Sep 24 2017 *)

Formula

a(n) = 3*6^n-3*2^n+1.

G.f.: (1+4*x)/((1-x)*(1-2*x)*(1-6*x)). [Colin Barker, May 08 2012]

Extensions

More terms from Karen Richardson (s1149414(AT)cedarville.edu)