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A178208
Number of ways to choose three points in an (n X n)-grid (or geoplane).
5
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
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
Sequence in context: A359861 A172138 A282588 * A069441 A203095 A006344
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