0,2

The lattice C_5 consists of all integer lattice points v = (x_1,...,x_5) in Z^5 such that (x_1 +...+ x_5) is even, equipped with the taxicab type norm ||v|| = 1/2 * (|x_1| +...+ |x_5|). The crystal ball sequence of C_5 gives the number of lattice points v in C_5 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

Partial sums of A019561. a(n) = (2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15 = sum {k = 0..5} C(10,2k)*C(n+k,5) = sum {k = 0..5} C(10,2k+1)*C(n+k+1/2,5). O.g.f.: (1+45*x+210*x^2+210*x^3+45*x^4+x^5)/(1-x)^6 = 1/(1-x) * T(5,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind. 2*log(2) = 41/30 + sum {n = 1..inf} 1/(n*a(n-1)*a(n)).

a(1) = 51. The origin has norm 0. The 50 lattice points in Z^5 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 5 and (+- e_i +- e_j), 1 <= i < j <= 5, where e_1,...,e_5 denotes the standard basis of Z^5. These 50 vectors form a root system of type C_5. Hence sequence begins 1, 1 + 50 = 51, ... .

a := n -> (2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15: seq(a(n), n = 0..20)

Cf. A019561, A063496, A142992, A142993.

Sequence in context: A083669 A164646 A128511 this_sequence A166820 A020278 A153221

Adjacent sequences: A142991 A142992 A142993 this_sequence A142995 A142996 A142997

easy,nonn

Peter Bala (pbala(AT)toucansurf.com), Jul 18 2008