A035606 Number of points of L1 norm 12 in cubic lattice Z^n.

0, 2, 48, 578, 4672, 28610, 142000, 596610, 2187520, 7159170, 21278640, 58227906, 148321344, 354870594, 803467056, 1732242690, 3575055360, 7094825730, 13591279920, 25216532290, 45443741760, 79749022402

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).
• M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
• Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Programs

Magma

I:=[0, 2, 48, 578, 4672, 28610, 142000, 596610, 2187520, 7159170, 21278640, 58227906, 148321344]; [n le 13 select I[n] else 13*Self(n-1)-78*Self(n-2)+286*Self(n-3)-715*Self(n-4)+1287*Self(n-5)-1716*Self(n-6)+1716*Self(n-7)-1287*Self(n-8)+715*Self(n-9)-286*Self(n-10)+78*Self(n-11)-13*Self(n-12)+Self(n-13): n in [1..30]]; // Vincenzo Librandi, Apr 24 2012

Maple

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

Mathematica

CoefficientList[Series[2*x*(1+x)^11/(1-x)^13,{x,0,30}],x](* Vincenzo Librandi, Apr 24 2012 *)
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,2,48,578,4672,28610,142000,596610,2187520,7159170,21278640,58227906,148321344},40] (* Harvey P. Dale, May 29 2021 *)

Formula

From Colin Barker, Apr 15 2012: (Start)

a(n) = 2*n^2*(146430 + 239327*n^2 + 74800*n^4 + 6996*n^6 + 220*n^8 + 2*n^10)/467775.

G.f.: 2*x*(1+x)^11/(1-x)^13. (End)