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

1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898

Offset

0,2

Comments

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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.

Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

Milan Janjić, Two Enumerative Functions.

Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000), 39-44.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(0)=1, a(n) = 16*n^2 + 2, n >= 1.

G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.

G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.

E.g.f.: (x*(x+1)*16+2)*e^x - 1. - Gopinath A. R., Feb 14 2012

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

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

From R. J. Mathar, May 07 2024: (Start)

Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth(Pi*sqrt(2)/4) = 1.095237238050...

a(n) = 2*A081585(n), n>0.

a(n) = A069129(n)+A069129(n+1). (End)

Sum_{n>=0} (-1)^n/a(n) = 3/4 + (sqrt(2)*Pi/16)*cosech(Pi*sqrt(2)/4). - Amiram Eldar, Nov 01 2025

Mathematica

Join[{1}, Table[16n^2+2, {n, 50}]] (* Harvey P. Dale, Oct 15 2012 *)

Prog

(PARI) A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012
(Magma) [1] cat [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

Crossrefs

Row n=3 of A103884.

Cf. A074377, A081585, A137513, 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).

Sequence in context: A327836 A165029 A264652 * A237616 A044156 A044537

Adjacent sequences: A010003 A010004 A010005 * A010007 A010008 A010009

Keyword

nonn,easy

Author

N. J. A. Sloane, Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

Status

approved