A035604 Number of points of L1 norm 10 in cubic lattice Z^n.

0, 2, 40, 402, 2720, 14002, 58728, 209762, 658048, 1854882, 4780008, 11414898, 25534368, 53972178, 108568488, 209070018, 387328512, 693230658, 1202893992, 2029779538, 3339504032, 5369283570, 8453107432, 13053926690, 19804348032, 29557550050, 43450388072

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• T. D. Noe, 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
• M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• 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 (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Crossrefs

Cf. A035607.

Programs

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

f[d_, m_] := Sum[2^i*Binomial[d, i]*Binomial[m-1, i-1], {i, 1, Min[d, m]}];
a[n_] := f[n, 10];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 24 2017, from Maple *)

PARI

x='x+O('x^99); concat(0, Vec(2*x*(1+x)^9/(1-x)^11)) \\ Altug Alkan, Mar 12 2018

Formula

a(n) = 2n^2/14175 * (2n^8 + 120n^6 + 1806n^4 + 7180n^2 + 5067).

G.f.: 2*x*(1+x)^9/(1-x)^11. - Colin Barker, Apr 15 2012

a(n) = 2*A099197(n). - R. J. Mathar, Dec 10 2013

a(n) = a(n-1) + A035603(n) + A035603(n-1). - Bruce J. Nicholson, Mar 11 2018