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A010006 Coordination sequence for C_3 lattice: a(n)=16*n^2+2 (n>0), a(0)=1. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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.

Milan Janjic, 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+15*x+15*x^2+x^3)/(1-x)^3.

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

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

a(0)=1, a(1)=18, a(2)=66, a(3)=146, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 15 2012

MATHEMATICA

t = Table[n^2 + (n + 1)^2, {n, 1, 100}]; Join[{1}, Table[t[[n]] + t[[n + 1]], {n, 1, Length[t] - 1, 2}]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

Join[{1}, Table[16n^2+2, {n, 50}]] (* or *) Join[{1}, LinearRecurrence[ {3, -3, 1}, {18, 66, 146}, 50]] (* Harvey P. Dale, Oct 15 2012 *)

PROG

(PARI) A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

(MAGMA) [1], [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

CROSSREFS

Cf. A206399.

Sequence in context: A259634 A165029 A264652 * A237616 A044156 A044537

Adjacent sequences:  A010003 A010004 A010005 * A010007 A010008 A010009

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified March 26 16:31 EDT 2017. Contains 284137 sequences.