A200048 - OEIS

%I #14 Nov 04 2017 17:55:11

%S 4,59,220,581,1162,2105,3370,5171,7384,10319,13744,18089,22990,29021,

%T 35662,43655,52300,62531,73444,86189,99634,115169,131410,150011,

%U 169312,191255,213880,239441,265654,295109,325174,358799,392980,431051,469612,512405,555610

%N Number of real singularities on a family of degree-3n algebraic surfaces.

%C The sequence gives the number of real nodes of a family of algebraic surfaces with degrees 3n. They have been introduced by means of a kind of duality in the basic geometric constructions corresponding to the generation of substitution tilings.

%H Colin Barker, <a href="/A200048/b200048.txt">Table of n, a(n) for n = 1..1000</a>

%H J. G. Escudero, <a href="http://arxiv.org/abs/1107.3401">A construction of algebraic surfaces with many real nodes</a>, arXiv:1107.3401

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

%F a(n) = (1/12)*(135*n^3-126*n^2 + 45*n-6) if n is odd; a(n) = (1/12)*(135*n^3- 117*n^2 + 54*n - 12) if n is even.

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

%F G.f.: x*(4 + 55*x + 149*x^2 + 196*x^3 + 110*x^4 + 25*x^5 + x^6) / ((1 - x)^4*(1 + x)^3).

%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7.

%F (End)

%o (PARI) a(n) = if (n % 2, (1/12)*(135*n^3-126*n^2 + 45*n-6), (1/12)*(135*n^3- 117*n^2 + 54*n - 12)); \\ _Michel Marcus_, Aug 17 2013

%o (PARI) Vec(x*(4 + 55*x + 149*x^2 + 196*x^3 + 110*x^4 + 25*x^5 + x^6) / ((1 - x)^4*(1 + x)^3) + O(x^40)) \\ _Colin Barker_, Nov 04 2017

%K nonn,easy

%O 1,1

%A _Juan G. Escudero_, Nov 13 2011

%E More terms from _Michel Marcus_, Aug 17 2013