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

0,3

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).

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A163956 A229661 A070724 * A022996 A023482 A273770

Adjacent sequences: A035875 A035876 A035877 * A035879 A035880 A035881

Keyword

nonn,easy

Author

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

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

Status

approved