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
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-François Alcover, Jul 14 2018, after Peter Luschny *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan G. Escudero, Nov 04 2017
STATUS
approved