

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
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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. 727762.
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), 11371142.


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)/(1x)^5 = 1/(1x) * T(4,(1+x)/(1x)), where T(n,x) denotes the Chebyshev polynomial of the first kind. 2*log(2) = 17/12  sum {n = 1..inf} 1/(n*a(n1)*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



