A210440 a(n) = 2*n*(n+1)*(n+2)/3.

0, 4, 16, 40, 80, 140, 224, 336, 480, 660, 880, 1144, 1456, 1820, 2240, 2720, 3264, 3876, 4560, 5320, 6160, 7084, 8096, 9200, 10400, 11700, 13104, 14616, 16240, 17980, 19840, 21824, 23936, 26180, 28560, 31080, 33744, 36556, 39520, 42640, 45920, 49364, 52976

Offset

0,2

Comments

Number of tin boxes necessary to build a tetrahedron with side length n, as shown in the link.

If "0" is prepended, a(n) is the convolution of 2n with itself. - Wesley Ivan Hurt, Mar 14 2015

a(n) is the number of ordered triples (y_1, y_2, y_3) of distinct elements drawn from a totally ordered set of size n+2 such that the triple is neither strictly increasing nor strictly decreasing. - Firdous Ahmad Mala, Jun 21 2026

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Pierre Gallais, Ceci n’est pas une mise en boîte !, Images des Mathématiques, CNRS, 2012.

Pierre Gallais, La vis ... sans fin, Images des Mathématiques, CNRS, 2012.

Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 29.

Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 4*A000292(n).

a(n+1)-a(n) = A046092(n+1).

From Bruno Berselli, Jan 20 2013: (Start)

G.f.: 4*x/(1-x)^4.

a(n) = -a(-n-2) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).

a(n)-a(-n) = A217873(n).

a(n)+a(-n) = A016742(n).

(n-1)*a(n)-n*a(n-1) = A130809(n+1) with n>1. (End)

From Bruno Berselli, Jan 21 2013: (Start)

a(n) = n*A028552(n) - Sum_{i=0..n-1} A028552(i) for n>0.

4*A001296(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n>0. (End)

G.f.: 2*x*W(0), where W(k) = 1 + 1/(1 - x*(k+2)*(k+4)/(x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 24 2013

a(n) = Sum_{i=1..n} i*(2*n-i+3). - Wesley Ivan Hurt, Oct 03 2013

From Amiram Eldar, Apr 30 2023: (Start)

Sum_{n>=1} 1/a(n) = 3/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) - 15/8. (End)

E.g.f.: 2*exp(x)*x*(6 + 6*x + x^2)/3. - Stefano Spezia, Jul 11 2025

Maple

A210440:=n->2*n*(n+1)*(n+2)/3; seq(A210440(k), k=0..100); # Wesley Ivan Hurt, Sep 25 2013

Mathematica

Table[2n(n+1)(n+2)/3, {n, 0, 50}] (* Harvey P. Dale, Feb 13 2013 *)
(* Alternative: *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 16, 40}, 50] (* Harvey P. Dale, Feb 13 2013 *)
(* Alternative: *)
CoefficientList[Series[4 x/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 24 2014 *)

Prog

(Maxima) A210440(n):=2*n*(n+1)*(n+2)/3$ makelist(A210440(n), n, 0, 20); /* Martin Ettl, Jan 22 2013 */
(Magma) [2*n*(n+1)*(n+2)/3: n in [0..50]]; // Vincenzo Librandi, Jun 24 2014

Crossrefs

Cf. A000292, A028552, A033488 (partial sums), A046092, A130809.

Sequence in context: A152133 A371345 A297361 * A329892 A220499 A331574

Adjacent sequences: A210437 A210438 A210439 * A210441 A210442 A210443

Keyword

nonn,easy

Author

Michel Marcus, Jan 20 2013

Status

approved