A225018 Number of cusps in a class of degree-3n complex algebraic surfaces.

3, 30, 127, 301, 647, 1100, 1851, 2715, 4027, 5434, 7463, 9545, 12447, 15336, 19267, 23095, 28211, 33110, 39567, 45669, 53623, 61060, 70667, 79571, 90987, 101490, 114871, 127105, 142607, 156704, 174483, 190575, 210787, 229006, 251807, 272285, 297831, 320700

Offset

1,1

Comments

The sequence gives the number of cusps of a family of algebraic surfaces with degrees 3n. They are obtained by using Belyi polynomials in combination with a class of complex polynomials related to the generation of surfaces with many ordinary double points.

Links

Table of n, a(n) for n=1..38.

J. G. Escudero, Hypersurfaces with many Aj-singularities: Explicit constructions, Journal of Computational and Applied Mathematics (2013)

J. G. Escudero, On a family of complex algebraic surfaces of degree 3n, 2013, arXiv:1302.6747 [math.AG]

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

Formula

a(n) = (1/2)*(12*n^3-9*n^2+4*n-1) if n is odd; a(n) = (1/2)*(12*n^3-12*n^2+7*n-2) if n is even.

G.f.: x*(3+27*x+88*x^2+93*x^3+64*x^4+12*x^5+x^6)/((1-x)^4*(1+x)^3). [Bruno Berselli, Apr 24 2013]

a(n) = (24*n^3-21*n^2+11*n-(3*n^2-3*n+1)*(-1)^n-3)/4. [Bruno Berselli, Apr 24 2013]

Mathematica

LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {3, 30, 127, 301, 647, 1100, 1851}, 40] (* Bruno Berselli, Apr 24 2013 *)

Prog

(Magma) [(24*n^3-21*n^2+11*n-(3*n^2-3*n+1)*(-1)^n-3)/4: n in [1..40]]; // Bruno Berselli, Apr 24 2013

Crossrefs

Sequence in context: A035328 A100259 A031205 * A365290 A221516 A174774

Adjacent sequences: A225015 A225016 A225017 * A225019 A225020 A225021

Keyword

nonn,easy

Author

Juan G. Escudero, Apr 24 2013

Status

approved