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Coordination sequence for C_4 lattice.
6

%I #28 May 28 2023 10:37:59

%S 1,32,192,608,1408,2720,4672,7392,11008,15648,21440,28512,36992,47008,

%T 58688,72160,87552,104992,124608,146528,170880,197792,227392,259808,

%U 295168,333600,375232,420192,468608

%N Coordination sequence for C_4 lattice.

%H Seiichi Manyama, <a href="/A019560/b019560.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)

%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.

%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://dx.doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

%H Joan Serra-Sagrista, <a href="http://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_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).

%F a(n) = (32/3)*n*(1 + 2*n^2) for n>0.

%F G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1 - x)^4.

%F G.f. for sequence with interpolated zeros: cosh(8*arctanh(x)) = 1/2*(((1 + x)/(1 - x))^4 + ((1 - x)/(1 + x))^4) = 1 + 32*x^2 + 192*x^4 + 608*x^6 + .... Cf. A057813. - _Peter Bala_, Apr 09 2017

%F a(n) = A008412(2*n). - _Seiichi Manyama_, Jun 08 2018

%t Join[{1}, Table[(32/3) n (1 + 2 n^2), {n, 30}]] (* _Vincenzo Librandi_, Apr 10 2017 *)

%o (Magma) [1] cat [(32/3)*n*(1 + 2*n^2): n in [1..40]]; // _Vincenzo Librandi_, Apr 10 2017

%Y Cf. A103884 (row 4). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

%Y Cf. A008412, A137513.

%K nonn,easy

%O 0,2

%A mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)