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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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).

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

Cf. A243211, A243208, A000217, A050534, A243213, A243214.

Sequence in context: A074877 A293263 A202255 * A232124 A232204 A232117

Adjacent sequences:  A243209 A243210 A243211 * A243213 A243214 A243215

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Jun 09 2014

STATUS

approved

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Last modified September 28 08:33 EDT 2020. Contains 337394 sequences. (Running on oeis4.)