A200048 Number of real singularities on a family of degree-3n algebraic surfaces.

4, 59, 220, 581, 1162, 2105, 3370, 5171, 7384, 10319, 13744, 18089, 22990, 29021, 35662, 43655, 52300, 62531, 73444, 86189, 99634, 115169, 131410, 150011, 169312, 191255, 213880, 239441, 265654, 295109, 325174, 358799, 392980, 431051, 469612, 512405, 555610

Offset

1,1

Comments

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

J. G. Escudero, A construction of algebraic surfaces with many real nodes, arXiv:1107.3401

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

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.

From Colin Barker, Nov 04 2017: (Start)

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).

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.

(End)

Prog

(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
(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

Crossrefs

Sequence in context: A198511 A282040 A245827 * A037066 A113251 A183462

Adjacent sequences: A200045 A200046 A200047 * A200049 A200050 A200051

Keyword

nonn,easy

Author

Juan G. Escudero, Nov 13 2011

Extensions

More terms from Michel Marcus, Aug 17 2013

Status

approved