A035877 Number of points of l_1 norm n in the "diamond" lattice D^+_2, i. e. the rectangular lattice generated by vectors (1, 1) and (-1/2, 1/2).

1, 2, 12, 6, 24, 10, 36, 14, 48, 18, 60, 22, 72, 26, 84, 30, 96, 34, 108, 38, 120, 42, 132, 46, 144, 50, 156, 54, 168, 58, 180, 62, 192, 66, 204, 70, 216, 74, 228, 78, 240, 82, 252, 86, 264, 90, 276, 94, 288, 98, 300, 102, 312, 106, 324, 110, 336, 114, 348, 118, 360, 122, 372, 126, 384, 130, 396

Offset: 0

Author

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

Keywords

• nonn
• easy

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-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, 2, 0, -1).

Crossrefs

Cf. A035878.

Programs

Maple

A035877 := proc(m) local k,t1; t1 := 2*binomial((2+2*m)/2-1,1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(2,k)*binomial(m-1,k-1),k=0..2); 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, 2], {m, 0, 40}] (* _Jean-François Alcover, Apr 18 2013, after Maple *)

Formula

a(n)*a(n+3) = -24 + a(n+1)*a(n+2).

G.f.: (1+2x+10x^2+2x^3+x^4)/(1-x^2)^2 and a(2n)=12n for n>0, a(2n+1)=4n+2.

Extensions

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

Name edited by Andrey Zabolotskiy, Aug 29 2022