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%I #60 Sep 08 2022 08:44:37
%S 1,18,66,146,258,402,578,786,1026,1298,1602,1938,2306,2706,3138,3602,
%T 4098,4626,5186,5778,6402,7058,7746,8466,9218,10002,10818,11666,12546,
%U 13458,14402,15378,16386,17426,18498,19602,20738,21906,23106,24338,25602,26898
%N Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.
%C 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 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 Vincenzo Librandi, <a href="/A010006/b010006.txt">Table of n, a(n) for n = 0..10000</a>
%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, 1997; 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%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_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
%F G.f.: (1 + 15*x + 15*x^2 + x^3)/(1-x)^3.
%F 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 E.g.f.: (x*(x+1)*16+2)*e^x - 1. - _Gopinath A. R._, Feb 14 2012
%F 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 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
%t Join[{1},Table[16n^2+2,{n,50}]] (* _Harvey P. Dale_, Oct 15 2012 *)
%o (PARI) A010006(n)=16*n^2+2-!n \\ _M. F. Hasler_, Feb 14 2012
%o (Magma) [1],[16*n^2+2: n in [1..50]]; // _Vincenzo Librandi_, Feb 20 2012
%Y 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 nonn,easy
%O 0,2
%A _N. J. A. Sloane_, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)
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