A035600 - OEIS

%I #43 Sep 08 2022 08:44:52

%S 0,2,24,146,608,1970,5336,12642,27008,53154,97880,170610,284000,

%T 454610,703640,1057730,1549824,2220098,3116952,4298066,5831520,

%U 7796978,10286936,13408034,17282432,22049250,27866072,34910514,43381856,53502738,65520920,79711106

%N Number of points of L1 norm 6 in cubic lattice Z^n.

%H Vincenzo Librandi, <a href="/A035600/b035600.txt">Table of n, a(n) for n = 0..1000</a>

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013

%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.

%H Joan Serra-Sagrista, <a href="https://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (4*n^6 + 40*n^4 + 46*n^2)/45. - _Frank Ellermann_, Mar 16 2002

%F G.f.: 2*x*(1+x)^5/(1-x)^7. - _Colin Barker_, Apr 15 2012

%F a(n) = 2*A069039(n). - _R. J. Mathar_, Dec 10 2013

%p 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

%t CoefficientList[Series[2*x*(1+x)^5/(1-x)^7,{x,0,33}],x] (* _Vincenzo Librandi_, Apr 23 2012 *)

%o (PARI) a(n)=(4*n^6+40*n^4+46*n^2)/45 \\ _Charles R Greathouse IV_, Dec 07 2011

%o (Magma)[( 4*n^6 +40*n^4 +46*n^2 )/45: n in [0..30]]; // _Vincenzo Librandi_, Apr 23 2012

%Y Cf. A035596-A035607.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_