A243212 Number of ways to place 3 points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.
0, 15, 107, 428, 1282, 3198, 7022, 14020, 26000, 45445, 75665, 120960, 186802, 280028, 409052, 584088, 817392, 1123515, 1519575, 2025540, 2664530, 3463130, 4451722, 5664828, 7141472, 8925553, 11066237, 13618360, 16642850, 20207160, 24385720, 29260400, 34920992
Offset: 2
Links
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).
Programs
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Magma
I:=[0,15,107,428,1282,3198,7022,14020]; [n le 8 select I[n] else 6*Self(n-1)-14*Self(n-2)+14*Self(n-3)-14*Self(n-5)+14*Self(n-6)-6*Self(n-7)+Self(n-8): n in [1..40]]; // Vincenzo Librandi, Jun 23 2015
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Mathematica
Table[Binomial[n (n + 1)/2, 3] - Floor[(n - 1) (n + 1) (2 n - 1)/8], {n, 2, 40}] (* Vincenzo Librandi, Jun 23 2015 *) -
PARI
concat(0, Vec(-x^3*(2*x^3-4*x^2+17*x+15)/((x-1)^7*(x+1)) + O(x^100))) \\ Colin Barker, Jun 09 2014
Formula
a(n) = C(n*(n+1)/2, 3) - floor((n-1)*(n+1)*(2*n-1)/8).
a(n) = C(n*(n+1)/2, 3) - A002717(n-1).
a(n) = (-3+3*(-1)^n+20*n+8*n^2-23*n^3-3*n^4+3*n^5+n^6)/48. - Colin Barker, Jun 09 2014
G.f.: -x^3*(2*x^3-4*x^2+17*x+15) / ((x-1)^7*(x+1)). - Colin Barker, Jun 09 2014