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%I #52 Dec 30 2023 13:28:24
%S 0,2,20,102,360,1002,2364,4942,9424,16722,28004,44726,68664,101946,
%T 147084,207006,285088,385186,511668,669446,864008,1101450,1388508,
%U 1732590,2141808,2625010,3191812,3852630,4618712,5502170,6516012
%N Number of points of L1 norm 5 in cubic lattice Z^n.
%H Vincenzo Librandi, <a href="/A035599/b035599.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. - _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_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = (4*n^4+20*n^2+6)*n/15. - _Frank Ellermann_, Mar 16 2002
%F G.f.: 2*x*(1+x)^4/(1-x)^6. - _Colin Barker_, Mar 19 2012
%F a(n) = 2*A069038(n). - _R. J. Mathar_, Dec 10 2013
%F From _Shel Kaphan_, Mar 01 2023: (Start)
%F a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=5.
%F a(n) = A001847(n) - A001846(n).
%F a(n) = A008413(n)*n/5. (End)
%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)^4/(1-x)^6,{x,0,33}],x] (* _Vincenzo Librandi_, Apr 23 2012 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,20,102,360,1002},40] (* _Harvey P. Dale_, Dec 30 2023 *)
%o (PARI) a(n)=(4*n^5+20*n^3+6*n)/15 \\ _Charles R Greathouse IV_, Dec 07 2011
%o (Magma) [(4*n^4+20*n^2+6)*n/15: n in [0..30]]; // _Vincenzo Librandi_, Apr 23 2012
%Y Cf. A035596-A035606.
%Y Column 5 of A035607, A266213. Row 5 of A113413, A119800, A122542.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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