A035878 Number of points of l_1 norm n in the "diamond" lattice D^+_4.

1, 0, 40, 32, 272, 160, 888, 448, 2080, 960, 4040, 1760, 6960, 2912, 11032, 4480, 16448, 6528, 23400, 9120, 32080, 12320, 42680, 16192, 55392, 20800, 70408, 26208, 87920, 32480, 108120, 39680, 131200, 47872, 157352, 57120, 186768, 67488, 219640, 79040, 256160

Offset: 0

Author

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

Keywords

• nonn
• easy

Examples

			This 4D lattice consists of points with coordinates that have even sum and are either all integer or all half-integer. (It is actually similar to Z^4.) The a(2) = 40 lattice vectors having l_1 norm 2 include: +-(1,1,1,1)/2, 6 permutations of (1,1,-1,-1)/2, 6 permutations with 4 choices of signs in (+-1,+-1,0,0), and 4 permutations with 2 choices of signs in (+-2,0,0,0), totaling 2 + 6 + 6*4 + 4*2 = 40.
		

Links

• Vincenzo Librandi, 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-Sagristà, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Crossrefs

Cf. A035877, A035879, A010369, A035881-A035926, A010006, A008412, A005925.

Programs

Maple

n := 4; A035878 := proc(m) global n; 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;

Mathematica

f[m_, n_] := 2^(n-1) *Binomial[(n + 2*m)/2 - 1, n - 1] + If[EvenQ[m], 2 *n* Hypergeometric2F1[1-m, 1-n, 2, 2], 0]; f[0, ] = 1; Table[f[m, 4], {m, 0, 32}] (* _Jean-François Alcover, Apr 18 2013, after Maple *)
CoefficientList[Series[(x^8 + 36 x^6 + 32 x^5 + 118 x^4 + 32 x^3 + 36 x^2 + 1)/((x - 1)^4 (x + 1)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2013 *)

Formula

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

G.f.: (x^8+36*x^6+32*x^5+118*x^4+32*x^3+36*x^2+1) / ((x-1)^4*(x+1)^4). - Colin Barker, Nov 18 2012

Extensions

Recomputed by N. J. A. Sloane, Nov 27 1998

More terms from Vincenzo Librandi, Oct 21 2013

Name edited by Andrey Zabolotskiy, Aug 29 2022