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

1,3

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., Vol. 13, Issue 4 (2010), Article 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.

Zhicong Lin, Feihu Liu, Jiahang Liu, Jing Liu, and Guoce Xin, Proof of a Conjecture on Young Tableaux with Walls, arXiv:2601.09551 [math.CO], 2026. See p. 32, Appendix B, Lemma B.7.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

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

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

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

From Amiram Eldar, Sep 28 2025: (Start)

Sum_{n>=3} 1/a(n) = 2/25.

Sum_{n>=3} (-1)^(n+1)/a(n) = 64*log(2)/5 - 661/75. (End)

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 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 15, 105, 420, 1260, 3150}, 40] (* Harvey P. Dale, Nov 05 2025 *)

Prog

(PARI) Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014
(Magma) [(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015
(PARI) vector(100, n, (n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015

Crossrefs

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

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

Sequence in context: A174385 A185129 A090454 * A256287 A047640 A296912

Adjacent sequences: A240437 A240438 A240439 * A240441 A240442 A240443

Keyword

nonn,easy

Author

Heinrich Ludwig, Apr 08 2014

Status

approved