A035878 -id:A035878 - cp's OEIS frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

1, 40, 272, 888, 2080, 4040, 6960, 11032, 16448, 23400, 32080, 42680, 55392, 70408, 87920, 108120, 131200, 157352, 186768, 219640, 256160, 296520, 340912, 389528, 442560, 500200, 562640, 630072, 702688, 780680, 864240, 953560, 1048832, 1150248, 1258000, 1372280

Offset: 0

N. J. A. Sloane, Apr 15 2008

This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Feb 05 2010

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.
Index entries for sequences related to D_4 lattice.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A035878, A105374, A110907, A144965, A175110.

I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4,{x,0,40}],x]  (* Vincenzo Librandi, Jun 27 2012 *)

From R. J. Mathar, Feb 03 2010, Feb 13 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - Colin Barker, May 24 2012
E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - Elmo R. Oliveira, Aug 18 2025

a(2) corrected and sequence extended by R. J. Mathar, Feb 03 2010, Feb 13 2010

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).

Original entry on oeis.org

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

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

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).

Cf. A035878.

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;

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 *)

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.

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

Name edited by Andrey Zabolotskiy, Aug 29 2022

cp's OEIS Frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

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