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A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

%I A010006 #71 Jan 23 2025 16:29:09
%S A010006 1,18,66,146,258,402,578,786,1026,1298,1602,1938,2306,2706,3138,3602,
%T A010006 4098,4626,5186,5778,6402,7058,7746,8466,9218,10002,10818,11666,12546,
%U A010006 13458,14402,15378,16386,17426,18498,19602,20738,21906,23106,24338,25602,26898
%N A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.
%C A010006 If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - _Milan Janjic_, Oct 28 2007
%C A010006 Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012
%H A010006 Vincenzo Librandi, <a href="/A010006/b010006.txt">Table of n, a(n) for n = 0..10000</a>
%H A010006 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, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
%H A010006 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 A010006 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A010006 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 A010006 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A010006 a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
%F A010006 G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.
%F A010006 G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
%F A010006 E.g.f.: (x*(x+1)*16+2)*e^x - 1. - _Gopinath A. R._, Feb 14 2012
%F A010006 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - _Harvey P. Dale_, Oct 15 2012
%F A010006 G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - _Peter Bala_, Apr 09 2017
%F A010006 Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth( Pi*sqrt(2)/4) = 1.095237238050... - _R. J. Mathar_, May 07 2024
%F A010006 a(n) = 2*A081585(n), n>0. - _R. J. Mathar_, May 07 2024
%F A010006 a(n) = A069129(n)+A069129(n+1). - _R. J. Mathar_, May 07 2024
%t A010006 Join[{1},Table[16n^2+2,{n,50}]] (* _Harvey P. Dale_, Oct 15 2012 *)
%o A010006 (PARI) A010006(n)=16*n^2+2-!n   \\ _M. F. Hasler_, Feb 14 2012
%o A010006 (Magma) [1] cat [16*n^2+2: n in [1..50]]; // _Vincenzo Librandi_, Feb 20 2012
%Y A010006 Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513.
%K A010006 nonn,easy
%O A010006 0,2
%A A010006 _N. J. A. Sloane_, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)