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A142993 Crystal ball sequence for the lattice C_4. 5
1, 33, 225, 833, 2241, 4961, 9633, 17025, 28033, 43681, 65121, 93633, 130625, 177633, 236321, 308481, 396033, 501025, 625633, 772161, 943041, 1140833, 1368225, 1628033, 1923201 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.].

LINKS

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 , p. 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.

FORMULA

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..inf} 1/(n*a(n-1)*a(n)).

EXAMPLE

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, ... .

MAPLE

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

CROSSREFS

Cf. A019560, A063496, A142992, A142994.

Sequence in context: A189180 A209531 A127870 * A230186 A075040 A274639

Adjacent sequences:  A142990 A142991 A142992 * A142994 A142995 A142996

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Jul 18 2008

STATUS

approved

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Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)