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A225018 Number of cusps in a class of degree-3n complex algebraic surfaces. 0
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 (list; graph; refs; listen; history; text; internal format)
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
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]
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
KEYWORD
nonn,easy
AUTHOR
Juan G. Escudero, Apr 24 2013
STATUS
approved

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Last modified March 4 02:53 EST 2024. Contains 370522 sequences. (Running on oeis4.)