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

2,3

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)

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A239123 A241697 A358994 * A219599 A221694 A229262

Adjacent sequences: A239566 A239567 A239568 * A239570 A239571 A239572

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 22 2014

Status

approved