A178208 Number of ways to choose three points in an (n X n)-grid (or geoplane).

0, 4, 84, 560, 2300, 7140, 18424, 41664, 85320, 161700, 287980, 487344, 790244, 1235780, 1873200, 2763520, 3981264, 5616324, 7775940, 10586800, 14197260, 18779684, 24532904, 31684800, 40495000, 51257700, 64304604, 80007984, 98783860, 121095300, 147455840, 178433024

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = A000938(n) + A045996(n).

a(n) = binomial(n^2,3) = 1/6*n^2*(n^2-1)*(n^2-2). - Martin Renner, May 23 2010

G.f.: 4*x^2*(1+x)*(1+13*x+x^2)/(1-x)^7. - Colin Barker, Jan 19 2012

a(1)=0, a(2)=4, a(3)=84, a(4)=560, a(5)=2300, a(6)=7140, a(7)=18424, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Nov 09 2012

Sum_{n>=2} 1/a(n) = Pi^2/2 - 15/4 - 3*Pi*cot(sqrt(2)*Pi)/(2*sqrt(2)). - Amiram Eldar, Feb 17 2024

Mathematica

Binomial[Range[30]^2, 3] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 4, 84, 560, 2300, 7140, 18424}, 30] (* Harvey P. Dale, Nov 09 2012 *)

Prog

(PARI) a(n)=binomial(n^2, 3) \\ Charles R Greathouse IV, Feb 19 2017

Crossrefs

Cf. A045996, A000938.

Sequence in context: A359861 A172138 A282588 * A069441 A203095 A006344

Adjacent sequences: A178205 A178206 A178207 * A178209 A178210 A178211

Keyword

easy,nice,nonn

Author

Martin Renner, May 22 2010

Extensions

Extended by Ray Chandler, May 03 2011

Corrected by Harvey P. Dale, Nov 09 2012

Status

approved