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.

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

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 A359928

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