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A239569
Number of ways to place 3 points on a triangular grid of side n so that no two of them are adjacent.
7
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.
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
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Mar 22 2014
STATUS
approved