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A035877 Number of points of l_1 norm n in the "diamond" lattice D^+_2, i. e. the rectangular lattice generated by vectors (1, 1) and (-1/2, 1/2).

%I A035877 #25 Sep 04 2023 10:30:42
%S A035877 1,2,12,6,24,10,36,14,48,18,60,22,72,26,84,30,96,34,108,38,120,42,132,
%T A035877 46,144,50,156,54,168,58,180,62,192,66,204,70,216,74,228,78,240,82,
%U A035877 252,86,264,90,276,94,288,98,300,102,312,106,324,110,336,114,348,118,360,122,372,126,384,130,396
%N A035877 Number of points of l_1 norm n in the "diamond" lattice D^+_2, i. e. the rectangular lattice generated by vectors (1, 1) and (-1/2, 1/2).
%H A035877 Ray Chandler, <a href="/A035877/b035877.txt">Table of n, a(n) for n = 0..1000</a>
%H A035877 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 A035877 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 A035877 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2, 0, -1).
%F A035877 a(n)*a(n+3) = -24 + a(n+1)*a(n+2).
%F A035877 G.f.: (1+2x+10x^2+2x^3+x^4)/(1-x^2)^2 and a(2n)=12n for n>0, a(2n+1)=4n+2.
%p A035877 A035877 := proc(m) local k,t1; t1 := 2*binomial((2+2*m)/2-1,1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(2,k)*binomial(m-1,k-1),k=0..2); fi; t1; end;
%t A035877 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, 2], {m, 0, 40}] (* _Jean-François Alcover_, Apr 18 2013, after Maple *)
%Y A035877 Cf. A035878.
%K A035877 nonn,easy
%O A035877 0,2
%A A035877 Joan Serra-Sagrista (jserra(AT)ccd.uab.es)
%E A035877 Recomputed by _N. J. A. Sloane_, Nov 27 1998
%E A035877 Name edited by _Andrey Zabolotskiy_, Aug 29 2022