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%I #33 Aug 29 2022 10:22:37
%S 1,0,40,32,272,160,888,448,2080,960,4040,1760,6960,2912,11032,4480,
%T 16448,6528,23400,9120,32080,12320,42680,16192,55392,20800,70408,
%U 26208,87920,32480,108120,39680,131200,47872,157352,57120,186768,67488,219640,79040,256160
%N Number of points of l_1 norm n in the "diamond" lattice D^+_4.
%H Vincenzo Librandi, <a href="/A035878/b035878.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 Joan Serra-Sagristà , <a href="https://doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F For n>0, a(n) = ( 2n^2 + 1 + (n^2+2)*(-1)^n ) * 4n/3.
%F G.f.: (x^8+36*x^6+32*x^5+118*x^4+32*x^3+36*x^2+1) / ((x-1)^4*(x+1)^4). - _Colin Barker_, Nov 18 2012
%e This 4D lattice consists of points with coordinates that have even sum and are either all integer or all half-integer. (It is actually similar to Z^4.) The a(2) = 40 lattice vectors having l_1 norm 2 include: +-(1,1,1,1)/2, 6 permutations of (1,1,-1,-1)/2, 6 permutations with 4 choices of signs in (+-1,+-1,0,0), and 4 permutations with 2 choices of signs in (+-2,0,0,0), totaling 2 + 6 + 6*4 + 4*2 = 40.
%p n := 4; A035878 := proc(m) global n; local k,t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1,n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n,k)*binomial(m-1,k-1),k=0..n); fi; t1; end;
%t 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, 4], {m, 0, 32}] (* _Jean-François Alcover_, Apr 18 2013, after Maple *)
%t CoefficientList[Series[(x^8 + 36 x^6 + 32 x^5 + 118 x^4 + 32 x^3 + 36 x^2 + 1)/((x - 1)^4 (x + 1)^4), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 21 2013 *)
%Y Cf. A035877, A035879, A010369, A035881-A035926, A010006, A008412, A005925.
%K nonn,easy
%O 0,3
%A Joan Serra-Sagrista (jserra(AT)ccd.uab.es)
%E Recomputed by _N. J. A. Sloane_, Nov 27 1998
%E More terms from _Vincenzo Librandi_, Oct 21 2013
%E Name edited by _Andrey Zabolotskiy_, Aug 29 2022
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