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A241219 Number of ways to choose two points on a centered hexagonal grid of size n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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. 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

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Last modified December 12 11:56 EST 2018. Contains 318060 sequences. (Running on oeis4.)