A010369 Number of points of L1 norm 2n in root system version of E_8 lattice.

1, 0, 128, 0, 2944, 1024, 31616, 15360, 199424, 101376, 877696, 439296, 3011200, 1464320, 8630144, 4073472, 21607936, 9922560, 48713856, 21829632, 101009792, 44301312, 195640192, 84198400, 358064384

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

Also, coordination sequence for diamond structure D^+8. (Edges defined by l_1 norm = 1.) - J. Serra-Sagrista (jserra(AT)ccd.uab.es). Confirmed by _N. J. A. Sloane Nov 27 1998.

Links

• Ray Chandler, Table of n, a(n) for n = 0..1000
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
• P. Solé, Counting lattice points in pyramids, Discr. Math. 139 (1995), 381-392.
• Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -28, 0, 56, 0, -70, 0, 56, 0, -28, 0, 8, 0, -1).

Crossrefs

Cf. A010368.

Programs

Maple

1/2*((1+z^2)^8+256*z^8)/(1-z^2)^8+1/2*(1-z^2)^8/(1+z^2)^8
f := proc(m) local k,t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1,n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n,k)*binomial(m-1,k-1),k=0..n); fi; t1; end; where n=8.

Formula

G.f.: (1/2)*((1+z^2)^8+256*z^8)/(1-z^2)^8 + (1/2)*(1-z^2)^8/(1+z^2)^8.