A030653 n^3*a(n) is the number of circles in complex projective plane tangent to three smooth curves of degree n in general position.

4, 23, 60, 121, 212, 339, 508, 725, 996, 1327, 1724, 2193, 2740, 3371, 4092, 4909, 5828, 6855, 7996, 9257, 10644, 12163, 13820, 15621, 17572, 19679, 21948, 24385, 26996, 29787, 32764, 35933, 39300, 42871, 46652, 50649, 54868, 59315

Offset

1,1

References

See formula for enumeration of contacts in Fulton-Kleiman-MacPherson (pp. 156-196 of Lect. Notes Math. n.997).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

William Fulton, Steven Kleiman, Robert MacPherson, About the enumeration of contacts, Algebraic Geometry — Open Problems, Lecture Notes in Mathematics, 1983 vol. 997.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = n^3 + 3*n^2 + 3*n - 3.

G.f.: x*(4 + 7*x - 8*x^2 + 3*x^3)/(1-x)^4. - Colin Barker, Sep 03 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 20 2022

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {4, 23, 60, 121}, 40] (* Harvey P. Dale, Nov 29 2014 *)

Prog

(Magma) [n^3+3*n^2+3*n-3: n in [1..45]]; // Vincenzo Librandi, Jun 30 2011
(PARI) a(n)=n^3 + 3*n^2 + 3*n - 3 \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A239624 A179628 A339392 * A085505 A065972 A132112

Adjacent sequences: A030650 A030651 A030652 * A030654 A030655 A030656

Keyword

nonn,nice,easy

Author

Paolo Dominici (pl.dm(AT)libero.it)

Status

approved