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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
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.
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
KEYWORD
easy,nonn
AUTHOR
Matthew Lehman, Mar 12 2009
EXTENSIONS
More terms from Michel Marcus, Aug 13 2013
STATUS
approved