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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.
5
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
2,2
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jun 09 2014
STATUS
approved