A239569 Number of ways to place 3 points on a triangular grid of side n so that no two of them are adjacent.

0, 1, 21, 151, 615, 1845, 4571, 9926, 19566, 35805, 61765, 101541, 160381, 244881, 363195, 525260, 743036, 1030761, 1405221, 1886035, 2495955, 3261181, 4211691, 5381586, 6809450, 8538725, 10618101, 13101921, 16050601, 19531065, 23617195, 28390296, 33939576

Offset: 2

Author

Heinrich Ludwig, Mar 22 2014

Keywords

• nonn
• easy

Comments

Rotations and reflections of placements are counted. If they are to be ignored, see A239573.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Crossrefs

Cf. A239567, A239573, A239568 (2 points), A239570 (4 points), A239571 (5 points), A282998 (6 points).

Programs

Magma

[(n^2-3*n+2)*(n^4+6*n^3-23*n^2-92*n+264)/48: n in [2..40]]; // Vincenzo Librandi, Mar 23 2014

Mathematica

CoefficientList[Series[- x (11 x^4 - 36 x^3 + 25 x^2 + 14 x + 1)/(x - 1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,21,151,615,1845,4571},50] (* Harvey P. Dale, Aug 08 2023 *)

PARI

concat(0, Vec(-x^3*(11*x^4-36*x^3+25*x^2+14*x+1)/(x-1)^7 + O(x^100))) \\ Colin Barker, Mar 22 2014

Formula

a(n) = (n-1)*(n-2)*(n^4+6*n^3-23*n^2-92*n+264)/48.

G.f.: -x^3*(11*x^4-36*x^3+25*x^2+14*x+1) / (x-1)^7. - Colin Barker, Mar 22 2014