A022145 Coordination sequence for root lattice B_3.

1, 18, 74, 170, 306, 482, 698, 954, 1250, 1586, 1962, 2378, 2834, 3330, 3866, 4442, 5058, 5714, 6410, 7146, 7922, 8738, 9594, 10490, 11426, 12402, 13418, 14474, 15570, 16706, 17882, 19098, 20354, 21650

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 74,..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Nov 02 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
• R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
• Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Programs

Magma

[1] cat [20*n^2-4*n+2: n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

Mathematica

CoefficientList[Series[(1+15*x+23*x^2+x^3)/(1-x)^3,{x,0,40}],x] (* Vincenzo Librandi, Apr 20 2012 *)
Join[{1},LinearRecurrence[{3,-3,1},{18,74,170},40]] (* Harvey P. Dale, Dec 03 2012 *)

Formula

a(n) = 20*n^2-4*n+2, for n>0.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. G.f.: (1+15*x+23*x^2+x^3)/(1-x)^3. [Colin Barker, Apr 13 2012]