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
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)
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*(2n-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)
MAPLE
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 *)
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
KEYWORD
nonn,easy,changed
AUTHOR
Michel Marcus, Jan 20 2013
STATUS
approved