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A294604 Number of ordinary double points of a family of threefolds. 0

%I

%S 10,41,120,283,566,1029,1738,2745,4150,6049,8504,11661,15646,20525,

%T 26496,33715,42246,52345,64198,77861,93654,111793,132320,155625,

%U 181954,211329,244216,280891,321350,366141,415570,469601,528870,593713,664056,740629,823798,913445,1010400,1115059,1227254

%N Number of ordinary double points of a family of threefolds.

%C 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.

%H J. G. Escudero, <a href="https://arxiv.org/abs/1107.3401">A construction of algebraic surfaces with many real nodes</a>, arXiv:1107.3401 [math-ph], 2011.

%H J. G. Escudero, <a href="https://doi.org/10.1007/s10231-015-0478-y">A construction of algebraic surfaces with many real nodes</a>, Annali di Matematica Pura ed Applicata, 195 (2016), 575-583.

%H J. G. Escudero, <a href="https://doi.org/10.1007/978-3-319-56932-1_7">The root lattice A2 in the construction of substitution tilings and singular hypersurfaces</a>, Springer Proceedings in Mathematics and Statistics, 198 (2017), 101-117.

%F 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, ...

%F Conjectures from _Colin Barker_, Nov 04 2017: (Start)

%F 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).

%F 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.

%F (End)

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

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

%p seq(a(n), n=3..43); # _Peter Luschny_, Nov 04 2017

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

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

%t Table[a[n], {n, 3, 43}] (* _Jean-Fran├žois Alcover_, Jul 14 2018, after _Peter Luschny_ *)

%Y Cf. A200048, A057870.

%K nonn

%O 3,1

%A _Juan G. Escudero_, Nov 04 2017

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Last modified August 15 01:47 EDT 2020. Contains 336485 sequences. (Running on oeis4.)