OFFSET
0,3
COMMENTS
The case n=1 is exceptional and a(1) could be 0 or 1.
A del Pezzo surface is a two-dimensional variety whose anticanonical divisor class is ample. The degree of a del Pezzo surface is the self-intersection of its canonical class. A del Pezzo surface of degree 9-n is isomorphic to the blow-up of the projective plane at n points in "general position", with one exception when n=1. An exceptional curve is a curve whose self-intersection is -1. These are also known as (-1)-curves. - Harry Richman, Mar 18 2025
a(n) is the number of vectors in the E8 lattice at minimal distance from a minimal regular simplex of dimension (8-n) consisting of (9-n) lattice vectors. They form the vertices of a uniform (n-4)_21 polytope for n >= 3, and the polytope is n-dimensional. In the case n = 2, the resulting triangle is isosceles but not regular. - Hal M. Switkay, Mar 26 2026
REFERENCES
Yu. I. Manin, Rational surfaces and Galois cohomology, pp. 495-509 of Proc. International Congress Mathematicians, Moscow 1966.
Yu. I. Manin, Cubic Forms, Second edition, North-Holland Publishing Co., Amsterdam, 1986, page 136, Theorem 26.2(iii), Table (IV.9).
LINKS
M. Nagata, On rational surfaces, I, Mem. Coll. Sci. Univ. Kyoto, Ser. A., XXXII (No. 3, 1960).
M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A., XXXIII (No. 2, 1960).
A. Neumaier, Lattices of simplex type, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 145--160. The sequence is on page 153.
Wikipedia, Del Pezzo surface
Wikipedia, Uniform k_21 polytope
EXAMPLE
G.f. = x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 27*x^6 + 56*x^7 + 240*x^8.
CROSSREFS
KEYWORD
fini,full,nonn
AUTHOR
N. J. A. Sloane, Feb 13 2002
STATUS
approved