0,2

The lattice C_4 consists of all integer lattice points v = (a,b,c,d) in Z^4 such that a + b + c + d is even, equipped with the taxicab type norm ||v|| = (1/2) * (|a| + |b| + |c| + |d|). The crystal ball sequence of C_4 gives the number of lattice points v in C_4 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

Seiichi Manyama, Table of n, a(n) for n = 0..10000

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.

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

Peter Bala, The C_4 lattice and a continued fraction for log(2)

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A019560.

a(n) = (2*n+1)^2*(4*n^2+4*n+3)/3 = Sum_{k = 0..4} C(8,2k)*C(n+k,4) = Sum_{k = 0..4} C(8,2k+1)*C(n+k+1/2,4).

O.g.f.: (1+28*x+70*x^2+28*x^3+x^4)/(1-x)^5 = (1/(1-x)) * T(4,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind.

2*log(2) = 17/12 - Sum_{n >= 1} 1/(n*a(n-1)*a(n)).

From Peter Bala, Mar 11 2024: (Start)

Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(33 - 3/(41 - 60/(57 - 315/(81 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*4^2))))).

E.g.f.: exp(x)*(1 + 32*x + 160*x^2/2! + 256*x^3/3! + 128*x^4/4!).

Note that T(8, i*sqrt(x)) = 1 + 32*x + 160*x^2 + 256*x^3 + 128*x^4. See A008310. (End)

a(1) = 33. The origin has norm 0. The 32 lattice points in Z^4 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 4 and (+- e_i +- e_j), 1 <= i < j <= 4, where e_1, e_2, e_3 and e_4 denotes the standard basis of Z^4. These 32 vectors form a root system of type C_4. Hence sequence begins 1, 1 + 32 = 33, ... .

a := n -> (2*n+1)^2*(4*n^2+4*n+3)/3: seq(a(n), n = 0..24)

Cf. A019560, A063496, A142992, A142994.

Sequence in context: A209531 A127870 A380069 * A230186 A075040 A353938

Adjacent sequences: A142990 A142991 A142992 * A142994 A142995 A142996

easy,nonn

Peter Bala, Jul 18 2008

approved