A240440 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.

0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880

Offset: 1

Author

Heinrich Ludwig, Apr 08 2014

Keywords

• nonn
• easy

Comments

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

a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

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

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

Programs

Magma

[(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015

Maple

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

Mathematica

Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)

PARI

Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014

PARI

vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015

Formula

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

G.f.: 15*x^3 / (1-x)^7. - Colin Barker, Apr 18 2014

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