A241219 Number of ways to choose two points on a centered hexagonal grid of size n.

0, 21, 171, 666, 1830, 4095, 8001, 14196, 23436, 36585, 54615, 78606, 109746, 149331, 198765, 259560, 333336, 421821, 526851, 650370, 794430, 961191, 1152921, 1371996, 1620900, 1902225, 2218671, 2573046, 2968266, 3407355, 3893445, 4429776, 5019696, 5666661

Offset: 1

Author

Martin Renner, Apr 17 2014

Keywords

• nonn
• easy

Comments

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

a(n) is also the number of segments on a centered hexagonal grid of size n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Hex Number.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A003215, A083374.

Programs

Magma

[Binomial(3*n^2-3*n+1, 2): n in [1..35]]; // Vincenzo Librandi, Apr 19 2014

Maple

seq(binomial(3*n^2-3*n+1, 2),n=1..34); # Martin Renner, Apr 27 2014
op(PolynomialTools[CoefficientList](convert(series(-3*x^2*(7*x^2+22*x+7)/(x-1)^5, x=0, 35), polynom), x)[2..35]); # Martin Renner, Apr 27 2014

Mathematica

CoefficientList[Series[-3 x^2 (7 x^2 + 22 x + 7)/(x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 19 2014 *)

PARI

concat(0, Vec(-3*x^2*(7*x^2+22*x+7) / (x-1)^5 + O(x^100))) \\ Colin Barker, Apr 18 2014

Formula

a(n) = binomial(A003215(n-1), 2).

= binomial(3*n^2-3*n+1, 2).

= 3/2*n*(n-1)*(3*n^2-3*n+1).

= 9/2*n^4-9*n^3+6*n^2-3/2*n.

G.f.: -3*x^2*(7*x^2+22*x+7) / (x-1)^5. - Colin Barker, Apr 18 2014

Sum_{n>=2} 1/a(n) = 8/3 - 2*Pi*tanh(Pi/(2*sqrt(3)))/sqrt(3). - Amiram Eldar, Feb 17 2024

Extensions

Typo in Mathematica program fixed by Martin Renner, Apr 27 2014