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A294604 Number of ordinary double points of a family of threefolds. 0
10, 41, 120, 283, 566, 1029, 1738, 2745, 4150, 6049, 8504, 11661, 15646, 20525, 26496, 33715, 42246, 52345, 64198, 77861, 93654, 111793, 132320, 155625, 181954, 211329, 244216, 280891, 321350, 366141, 415570, 469601, 528870, 593713, 664056, 740629, 823798, 913445, 1010400, 1115059, 1227254 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

The degree-n projective algebraic threefolds have been obtained from a class of polynomials introduced for the construction of nodal surfaces. The threefolds have ordinary double points as their only singularities.

LINKS

Table of n, a(n) for n=3..43.

J. G. Escudero, A construction of algebraic surfaces with many real nodes, arXiv:1107.3401 [math-ph], 2011.

J. G. Escudero, A construction of algebraic surfaces with many real nodes, Annali di Matematica Pura ed Applicata, 195 (2016), 575-583.

J. G. Escudero, The root lattice A2 in the construction of substitution tilings and singular hypersurfaces, Springer Proceedings in Mathematics and Statistics, 198 (2017), 101-117.

FORMULA

a(n) = (1/18)*(7*n^4 - 24*n^3 + 39*n^2 - 36*n + 18) if n is divisible by 3; a(n) = (1/18)*(7*n^4 - 24*n^3 + 37*n^2 - 30*n + 10) otherwise. For n = 3, 4, 5, ...

Conjectures from Colin Barker, Nov 04 2017: (Start)

G.f.: x^3*(10 + 21*x + 48*x^2 + 54*x^3 + 57*x^4 + 36*x^5 + 24*x^6 + x^7 + 2*x^8 - 2*x^9 + x^10) / ((1 - x)^5*(1 + x + x^2)^3).

a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-5) - 3*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) - 2*a(n-10) + a(n-11) for n > 10.

(End)

MAPLE

alpha := n -> (7*n^4-24*n^3+39*n^2-36*n+18)/18:

a := n -> `if`(modp(n, 3)=0, alpha(n), alpha(n)-((n-2)^2+n)/9):

seq(a(n), n=3..43); # Peter Luschny, Nov 04 2017

MATHEMATICA

alpha[n_] := (7*n^4 - 24*n^3 + 39*n^2 - 36*n + 18)/18;

a[n_] := If[Mod[n, 3] == 0, alpha[n], alpha[n] - ((n-2)^2 + n)/9];

Table[a[n], {n, 3, 43}] (* Jean-Fran├žois Alcover, Jul 14 2018, after Peter Luschny *)

CROSSREFS

Cf. A200048, A057870.

Sequence in context: A006323 A178073 A102784 * A061003 A211064 A048879

Adjacent sequences:  A294601 A294602 A294603 * A294605 A294606 A294607

KEYWORD

nonn

AUTHOR

Juan G. Escudero, Nov 04 2017

STATUS

approved

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Last modified July 11 08:50 EDT 2020. Contains 335626 sequences. (Running on oeis4.)