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

Author

Michel Marcus, Jan 20 2013

Keywords

• nonn
• easy

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

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).

Crossrefs

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

Programs

Magma

[2*n*(n+1)*(n+2)/3: n in [0..50]]; // Vincenzo Librandi, Jun 24 2014

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

Maxima

A210440(n):=2*n*(n+1)*(n+2)/3$ makelist(A210440(n),n,0,20); /* Martin Ettl, Jan 22 2013 */

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