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%I A158121 #25 Sep 08 2022 08:45:42
%S A158121 6,93,591,2381,7316,18761,42253,86281,163186,290181,490491,794613,
%T A158121 1241696,1881041,2773721,3994321,5632798,7796461,10612071,14228061,
%U A158121 18816876,24577433,31737701,40557401,51330826,64389781,80106643,98897541
%N A158121 Given n points in the complex plane, let M(n) the number of distinct Moebius transformations that take 3 distinct points to 3 distinct points. Note that the triples may have some or all of the points in common.
%C A158121 There are (nC3)^2 ways of choosing two triples out of n points with repetition.
%C A158121 There are 3! = 6 ways of mapping the points of one triple to the other.
%C A158121 However, given each triple pair, there is one case where each of the initial three points is mapped to itself, resulting in the identity Moebius transformation.
%C A158121 There are nC3 cases of this, all but one redundant.
%D A158121 Michael P. Hitchman, Geometry With an Introduction to Cosmic Topology, Jones and Bartlett Publishers, 2009, pages 59-60.
%H A158121 Vincenzo Librandi, <a href="/A158121/b158121.txt">Table of n, a(n) for n = 3..1000</a>
%H A158121 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A158121 M(n) = 6*C(n,3)^2 - C(n,3) + 1.
%F A158121 M(n) = 1/6*(n^6-6*n^5+13*n^4-13*n^3+7*n^2-2*n+6).
%F A158121 G.f.: x^3*(6+51*x+66*x^2-13*x^3+15*x^4-6*x^5+x^6)/(1-x)^7. - _Colin Barker_, May 02 2012
%e A158121 For n=3, M(3) = 3! = 6, since there aren't any redundancies.
%e A158121 For n=4, M(4) = (6*4^2) - 3 = 93, since there are 3 redundant mappings.
%t A158121 CoefficientList[Series[(6 + 51 x + 66 x^2 - 13 x^3 + 15 x^4 - 6 x^5 + x^6) / (1 - x)^7, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 14 2013 *)
%t A158121 LinearRecurrence[{7,-21,35,-35,21,-7,1},{6,93,591,2381,7316,18761,42253},30] (* _Harvey P. Dale_, Mar 07 2020 *)
%o A158121 (PARI) a(n) = 6* binomial(n, 3)^2 - binomial(n, 3) + 1; \\ _Michel Marcus_, Aug 13 2013
%o A158121 (Magma) I:=[6,93,591,2381,7316,18761,42253]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..30]]; // _Vincenzo Librandi_, Aug 14 2013
%K A158121 easy,nonn
%O A158121 3,1
%A A158121 _Matthew Lehman_, Mar 12 2009
%E A158121 More terms from _Michel Marcus_, Aug 13 2013

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