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A210440 a(n) = 2*n*(n+1)*(n+2)/3. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

This sequence is related to A028552 by a(n) = n*A028552(n)-sum(A028552(i), i=0..n-1) with n>0. Also, 4*A001296(n) = n*a(n)-sum(a(i), i=0..n-1) with n>0. [Bruno Berselli, Jan 21 2013]

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

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

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

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)

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

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 *)

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: A121318 A152133 A297361 * A220499 A110477 A007057

Adjacent sequences:  A210437 A210438 A210439 * A210441 A210442 A210443

KEYWORD

nonn,easy

AUTHOR

Michel Marcus, Jan 20 2013

STATUS

approved

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Last modified May 19 16:48 EDT 2019. Contains 323395 sequences. (Running on oeis4.)