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Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.
6

%I #39 Jul 06 2024 10:25:34

%S 0,0,15,105,420,1260,3150,6930,13860,25740,45045,75075,120120,185640,

%T 278460,406980,581400,813960,1119195,1514205,2018940,2656500,3453450,

%U 4440150,5651100,7125300,8906625,11044215,13592880,16613520,20173560,24347400,29216880

%N Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

%C a(n) = 15 * A000579(n+3).

%C a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - _Ran Pan_, Dec 03 2015

%H Vincenzo Librandi, <a href="/A240440/b240440.txt">Table of n, a(n) for n = 1..1000</a>

%H Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48.

%F G.f.: 15*x^3 / (1-x)^7. - _Colin Barker_, Apr 18 2014

%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7. - _Wesley Ivan Hurt_, Dec 03 2015

%p A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # _Wesley Ivan Hurt_, Apr 08 2014

%t Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* _Wesley Ivan Hurt_, Apr 08 2014 *)

%t CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 19 2014 *)

%o (PARI) Vec(15*x^3/(1-x)^7 + O(x^100)) \\ _Colin Barker_, Apr 18 2014

%o (Magma) [(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // _Wesley Ivan Hurt_, Dec 03 2015

%o (PARI) vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ _Derek Orr_, Dec 24 2015

%Y Cf. A000579, A001498, A240441, A240442, A240439.

%Y If one of the initial zeros is omitted, this is a row of the array in A129533.

%K nonn,easy

%O 1,3

%A _Heinrich Ludwig_, Apr 08 2014