A240826 Number of ways to choose three points on a centered hexagonal grid of size n.

0, 35, 969, 7770, 35990, 121485, 333375, 790244, 1679580, 3280455, 5989445, 10349790, 17083794, 27128465, 41674395, 62207880, 90556280, 128936619, 180007425, 246923810, 333395790, 443749845, 582993719, 756884460, 971999700, 1235812175, 1556767485, 1944365094

Offset

1,2

Comments

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

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 (7,-21,35,-35,21,-7,1).

Formula

a(n) = binomial(A003215(n-1), 3)

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

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

= 9/2*n^6-27/2*n^5+27/2*n^4-9/2*n^3-1/2*n^2+1/2*n.

G.f.: -x^2*(35*x^4+724*x^3+1722*x^2+724*x+35) / (x-1)^7. - Colin Barker, Apr 18 2014

Sum_{n>=2} 1/a(n) = sqrt(3/7)*Pi*tan(sqrt(7/3)*Pi/2) + sqrt(3)*Pi*tanh(Pi/(2*sqrt(3))) - 2. - Amiram Eldar, Feb 17 2024

Maple

seq(binomial(3*n^2-3*n+1, 3), n=1..28); # Martin Renner, May 31 2014
op(PolynomialTools[CoefficientList](convert(series(-x^2*(35*x^4+724*x^3+1722*x^2+724*x+35)/(x-1)^7, x=0, 29), polynom), x)[2..29]); # Martin Renner, May 31 2014

Mathematica

CoefficientList[Series[- x(35 x^4 + 724 x^3 + 1722 x^2 + 724 x + 35)/(x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 35, 969, 7770, 35990, 121485, 333375}, 40] (* Harvey P. Dale, Sep 12 2019 *)

Crossrefs

Cf. A003215, A178208, A241219.

Sequence in context: A014934 A115473 A002453 * A210313 A049395 A278723

Adjacent sequences: A240823 A240824 A240825 * A240827 A240828 A240829

Keyword

nonn,easy

Author

Martin Renner, Apr 17 2014

Status

approved