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 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. 1
 6, 93, 591, 2381, 7316, 18761, 42253, 86281, 163186, 290181, 490491, 794613, 1241696, 1881041, 2773721, 3994321, 5632798, 7796461, 10612071, 14228061, 18816876, 24577433, 31737701, 40557401, 51330826, 64389781, 80106643, 98897541 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS There are (nC3)^2 ways of choosing two triples out of n points with repetition. There are 3! = 6 ways of mapping the points of one triple to the other. 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. There are nC3 cases of this, all but one redundant. REFERENCES Michael P. Hitchman, Geometry With an Introduction to Cosmic Topology, Jones and Bartlett Publishers, 2009, pages 59-60. LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..1000 Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). FORMULA M(n) = 6*C(n,3)^2 - C(n,3) + 1. M(n) = 1/6*(n^6-6*n^5+13*n^4-13*n^3+7*n^2-2*n+6). 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 EXAMPLE For n=3, M(3) = 3! = 6, since there aren't any redundancies. For n=4, M(4) = (6*4^2) - 3 = 93, since there are 3 redundant mappings. MATHEMATICA 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 *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {6, 93, 591, 2381, 7316, 18761, 42253}, 30] (* Harvey P. Dale, Mar 07 2020 *) PROG (PARI) a(n) = 6* binomial(n, 3)^2 - binomial(n, 3) + 1; \\ Michel Marcus, Aug 13 2013 (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 CROSSREFS Sequence in context: A009527 A053512 A331623 * A328427 A103212 A033935 Adjacent sequences:  A158118 A158119 A158120 * A158122 A158123 A158124 KEYWORD easy,nonn AUTHOR Matthew Lehman, Mar 12 2009 EXTENSIONS More terms from Michel Marcus, Aug 13 2013 STATUS approved

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Last modified September 22 12:34 EDT 2020. Contains 337289 sequences. (Running on oeis4.)