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

Author

Matthew Lehman, Mar 12 2009

Keywords

• easy
• nonn

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.

Examples

			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.
		

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).

Programs

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

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 *)

PARI

a(n) = 6* binomial(n, 3)^2 - binomial(n, 3) + 1; \\ Michel Marcus, Aug 13 2013

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

Extensions

More terms from Michel Marcus, Aug 13 2013