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

%I #77 Oct 19 2024 15:57:32

%S 0,4,16,40,80,140,224,336,480,660,880,1144,1456,1820,2240,2720,3264,

%T 3876,4560,5320,6160,7084,8096,9200,10400,11700,13104,14616,16240,

%U 17980,19840,21824,23936,26180,28560,31080,33744,36556,39520,42640,45920,49364,52976

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

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

%C If "0" is prepended, a(n) is the convolution of 2n with itself. - _Wesley Ivan Hurt_, Mar 14 2015

%H Vincenzo Librandi, <a href="/A210440/b210440.txt">Table of n, a(n) for n = 0..1000</a>

%H Pierre Gallais, <a href="http://images-archive.math.cnrs.fr/Ceci-n-est-pas-une-mise-en-boite.html">Ceci n’est pas une mise en boîte !</a>, Images des Mathématiques, CNRS, 2012.

%H Pierre Gallais, <a href="http://images-archive.math.cnrs.fr/La-vis-sans-fin.html">La vis ... sans fin</a>, Images des Mathématiques, CNRS, 2012.

%H Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, <a href="https://arxiv.org/abs/2307.13749">Semi-simplicial combinatorics of cyclinders and subdivisions</a>, arXiv:2307.13749 [math.CO], 2023. See p. 29.

%H Pakawut Jiradilok, <a href="https://arxiv.org/abs/2404.02714">Some Combinatorial Formulas Related to Diagonal Ramsey Numbers</a>, arXiv:2404.02714 [math.CO], 2024. See p. 19.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

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

%F From _Bruno Berselli_, Jan 20 2013: (Start)

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

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

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

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

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

%F From _Bruno Berselli_, Jan 21 2013: (Start)

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

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

%F 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

%F a(n) = Sum_{i=1..n} i*(2n-i+3). - _Wesley Ivan Hurt_, Oct 03 2013

%F From _Amiram Eldar_, Apr 30 2023: (Start)

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

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

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

%t 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 *)

%t CoefficientList[Series[4 x/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 24 2014 *)

%o (Maxima) A210440(n):=2*n*(n+1)*(n+2)/3$ makelist(A210440(n),n,0,20); /* _Martin Ettl_, Jan 22 2013 */

%o (Magma) [2*n*(n+1)*(n+2)/3: n in [0..50]]; // _Vincenzo Librandi_, Jun 24 2014

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

%K nonn,easy

%O 0,2

%A _Michel Marcus_, Jan 20 2013