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

1,2

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).

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

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 *)

Prog

(PARI) concat(0, Vec(-3*x^2*(7*x^2+22*x+7) / (x-1)^5 + O(x^100))) \\ Colin Barker, Apr 18 2014
(Magma) [Binomial(3*n^2-3*n+1, 2): n in [1..35]]; // Vincenzo Librandi, Apr 19 2014

Crossrefs

Cf. A003215, A083374.

Sequence in context: A041848 A125358 A126516 * A185128 A007261 A119105

Adjacent sequences: A241216 A241217 A241218 * A241220 A241221 A241222

Keyword

nonn,easy

Author

Martin Renner, Apr 17 2014

Extensions

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

Status

approved