%I #29 Mar 03 2024 11:10:10

%S 0,0,4,34,220,1330,7924,47194,281740,1685410,10095844,60522154,

%T 362968060,2177301490,13062263764,78368897914,470199235180,

%U 2821152757570,16926788191684,101560343302474,609360900699100

%N Number of subgroups of index 3 in fundamental group of a closed surface of characteristic -n.

%D 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.

%H Vincenzo Librandi, <a href="/A049293/b049293.txt">Table of n, a(n) for n = -2..1000</a>

%H V. A. Liskovets and A. Mednykh, <a href="https://www.researchgate.net/publication/251203042">Number of non-orientable coverings of the Klein bottle</a>, on ResearchGate.

%H A. D. Mednykh, <a href="https://doi.org/10.1080/00927878808823684">On the number of subgroups in the fundamental group of a closed surface</a>, Commun. in Algebra, 16, No 10 (1988), 2137-2148.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-47,72,-36).

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

%F a(-2)=0, a(-1)=0, a(0)=4, a(1)=34, a(n)=12*a(n-1)-47*a(n-2)+72*a(n-3)- 36*a(n-4) [_Harvey P. Dale_, Mar 03 2012]

%F G.f.: 2*(2-7*x)/((1-x)*(1-2*x)*(1-3*x)*(1-6*x)). - _Bruno Berselli_, Mar 04 2012

%t Table[6^(n+1)+3^(n+1)-3*2^(n+1)+1,{n,-2,20}] (* or *) LinearRecurrence[ {12,-47,72,-36},{0,0,4,34},30] (* _Harvey P. Dale_, Mar 03 2012 *)

%o (PARI) a(n)=6^(n+1)+3^(n+1)-3<<(n+1)+1 \\ _Charles R Greathouse IV_, Mar 04, 2012

%Y Cf. A003319, A027837, A049290-A049295.

%K nonn,easy,nice

%O -2,3

%A _Valery A. Liskovets_

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

%E Corrected by _T. D. Noe_, Nov 08 2006