A047786 a(n) = (9*n^4 + 4*n^3 - n)/2.

0, 6, 87, 417, 1278, 3060, 6261, 11487, 19452, 30978, 46995, 68541, 96762, 132912, 178353, 234555, 303096, 385662, 484047, 600153, 735990, 893676, 1075437, 1283607, 1520628, 1789050, 2091531, 2430837, 2809842, 3231528, 3698985, 4215411, 4784112, 5408502

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

In a triangular lattice, draw a regular hexagon of side length n (that is, n+1 points on a side). Then a(n) is the number of ways to choose two lattice points that do not lie on a line parallel to any side of the hexagon. (See the Bennett and Potts paper.) - Mikhail Lavrov, Jun 12 2023

Examples

			From _Mikhail Lavrov_, Jun 12 2023: (Start)
For n=1 the a(1)=6 ways to choose two points are illustrated below:
.
.    X o     o X     o o     o o     X o     o X
.   o o X   o o o   o o X   X o o   o o o   X o o
.    o o     o X     X o     o X     X o     o o
.
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31 (see p. 30).
• B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

The number of points in the hexagon is A003215. The number of ways to choose 2n+1 points, no two of which are on a line, is A002047.

Programs

GAP

List([0..30], n-> n*(9*n^3 +4*n^2 -1)/2) # G. C. Greubel, May 17 2019

Magma

[(9*n^4+4*n^3-n)/2: n in [0..40]]; // Vincenzo Librandi, May 29 2016

Mathematica

Table[(9n^4+4n^3-n)/2,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,6,87,417,1278},30] (* Harvey P. Dale, May 26 2016 *)

PARI

{a(n) = n*(9*n^3 +4*n^2 -1)/2}; \\ G. C. Greubel, May 17 2019

Sage

[n*(9*n^3 +4*n^2 -1)/2 for n in (0..30)] # G. C. Greubel, May 17 2019

Formula

O.g.f.: 3*x*(2 + 19*x + 14*x^2 + x^3)/(1-x)^5. - R. J. Mathar, Feb 26 2008

E.g.f.: x*(12 + 75*x + 58*x^2 + 9*x^3)*exp(x)/2. - Robert Israel, May 29 2016