A178208 - OEIS

%I #31 Feb 17 2024 04:04:05

%S 0,4,84,560,2300,7140,18424,41664,85320,161700,287980,487344,790244,

%T 1235780,1873200,2763520,3981264,5616324,7775940,10586800,14197260,

%U 18779684,24532904,31684800,40495000,51257700,64304604,80007984,98783860,121095300,147455840,178433024

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

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

%F a(n) = binomial(n^2,3) = 1/6*n^2*(n^2-1)*(n^2-2). - _Martin Renner_, May 23 2010

%F G.f.: 4*x^2*(1+x)*(1+13*x+x^2)/(1-x)^7. - _Colin Barker_, Jan 19 2012

%F 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

%F 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

%t 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 *)

%o (PARI) a(n)=binomial(n^2,3) \\ _Charles R Greathouse IV_, Feb 19 2017

%Y Cf. A045996, A000938.

%K easy,nice,nonn

%O 1,2

%A _Martin Renner_, May 22 2010

%E Extended by _Ray Chandler_, May 03 2011

%E Corrected by _Harvey P. Dale_, Nov 09 2012