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
Links
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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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 *) -
PARI
a(n)=binomial(n^2,3) \\ Charles R Greathouse IV, Feb 19 2017
Formula
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
Extensions
Extended by Ray Chandler, May 03 2011
Corrected by Harvey P. Dale, Nov 09 2012