1,2

Number of potential flows in a 2 X 2 matrix with integer velocities in -n..n, i.e. number of 2 X 2 matrices with adjacent elements differing by no more than n, counting matrices differing by a constant only once. - Ron Hardin (rhh(AT)cadence.com), Feb 27 2002

Number of ordered quadruples (a,b,c,d), -(n-1)<= a,b,c,d<=n-1, such that a+b+c+d=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003

If Y and Z are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Oct 28 2007

Equals binomial transform of [1, 18, 48, 32, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).

Milan Janjic, Two Enumerative Functions

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.

Comments from Peter Bala (pbala@toucansurf.com), Jul 18 2008 (Start): Partial sums of A010006. So this sequence is the crystal ball sequence for the C_3 lattice – row 3 of A142992. The lattice C_3 consists of all integer lattice points v = (a,b,c) in Z^3 such that a + b + c is even, equipped with the taxicab type norm ||v|| = 1/2 * (|a| + |b| + |c|).

The crystal ball sequence of C_3 gives the number of lattice points v in C_3 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

For example, a(1) = 19 because the origin has norm 0 and the 18 lattice points in Z^3 of norm 1 (as defined above) are +-(2,0,0), +-(0,2,0), +-(0,0,2), +-(1,1,0), +-(1,0,1), +-(0,1,1), +-(1,-1,0), +-(1,0,-1) and +-(0,1,-1). These 18 vectors form a root system of type C_3.

O.g.f.: x*(1+15*x+15*x^2+x^3)/(1-x)^4 = x/(1-x) * T(3,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind.

2*log(2) = 4/3 + sum {n = 1..inf} 1/(n*a(n)*a(n+1)). (End)

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A003215.

Sequence in context: A036564 A062639 A039609 this_sequence A027848 A039454 A142089

Adjacent sequences: A063493 A063494 A063495 this_sequence A063497 A063498 A063499

Cf. A010006, A142992, A142993, A142994 .

nonn

njas, Aug 01 2001