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%I #25 Jun 30 2017 03:02:05
%S 1,71,2143,54335,1321471,31817471,764217343,18344733695,440294408191,
%T 10567189327871,253613279903743,6086723113107455,146081381003558911,
%U 3505953301484470271,84142880178680889343,2019429129941297135615,48466299152487396933631
%N Number of subgroups of index 4 in free group of rank n+1.
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23, N_{4,n}.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
%H M. Hall, <a href="http://dx.doi.org/10.4153/CJM-1949-017-2">Subgroups of finite index in free groups</a>, Canad. J. Math., 1 (1949), 187-190.
%H V. A. Liskovets and A. Mednykh, <a href="http://dx.doi.org/10.1080/00927870008826924">Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces</a>, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (37,-368,1436,-2256,1152).
%F a(n) = 4*24^n-4*6^n-2*4^n+4*2^n-1.
%F G.f.: (264*x^3+116*x^2-34*x-1) / ((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(24*x-1)). [_Colin Barker_, Feb 17 2013]
%t LinearRecurrence[{37,-368,1436,-2256,1152},{1,71,2143,54335,1321471},20] (* _Harvey P. Dale_, Apr 14 2016 *)
%Y Cf. A003319, A027837, A049290-A049294.
%K nonn,easy,nice
%O 0,2
%A _Valery A. Liskovets_
%E More terms from Carrie Westbrook (s1213407(AT)cedarville.edu)
%E Terms corrected by _Colin Barker_, May 08 2012
%E a(16) from _Colin Barker_, Feb 17 2013
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