A059986 Number of rods required to make a 3-D cube of side length n.

0, 12, 54, 144, 300, 540, 882, 1344, 1944, 2700, 3630, 4752, 6084, 7644, 9450, 11520, 13872, 16524, 19494, 22800, 26460, 30492, 34914, 39744, 45000, 50700, 56862, 63504, 70644, 78300, 86490, 95232, 104544, 114444, 124950, 136080, 147852, 160284, 173394

Offset

0,2

Comments

Equals number of rods making a cube of side length n+1 minus the number of line segments illustrating the isometric projection of a cube of side length n+1 (i.e., the hexagonal matchstick numbers). See the illustration in the links and formula below. - Peter M. Chema, Mar 14 2017

a(n) is also the edge count and intersection number of the (n+1) X (n+1) X (n+1) grid graph. - Eric W. Weisstein, Mar 09 2024

Links

Table of n, a(n) for n=0..38.

Peter M. Chema, First difference are the hexagonal matchstick numbers or isometric projection of a cube.

Eric Weisstein's World of Mathematics, Edge Count.

Eric Weisstein's World of Mathematics, Grid Graph.

Eric Weisstein's World of Mathematics, Intersection Number.

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

Formula

a(n) = 3*n*(n+1)^2. - Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005

From Geoffrey Critzer, May 17 2009: (Start)

a(n) = a(n-1) + 9*n^2 + 3*n.

O.g.f.: 6*x*(2 + x)/(1 - x)^4.

E.g.f.: 3*x*exp(x)*(x^2 + 5*x + 4). (End)

a(n) = A117227(n^3). - Michel Marcus, Jun 19 2013

For n > 0, a(n) = Sum_{k=1..n} 2*(n+1)(k+n+1), which is the sum of all perimeters of Pythagorean triangles with arms 2*k*(n+1) and (n+1)^2 - k^2 with hypotenuse k^2 + (n+1)^2. - J. M. Bergot, May 12 2014

a(n) = a(n+1) - A045945(n+1). - Peter M. Chema, Mar 14 2017

a(n) = (n-1)*t(n+1) + n*(t(n)+t(n+1)) + (n+1)*(t(n-1)+t(n)+t(n+1)), where t = A000217. - J. M. Bergot, May 30 2017

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=1} 1/a(n) = 2/3 - Pi^2/18.

Sum_{n>=1} (-1)^(n+1)/a(n) = -2/3 + Pi^2/36 + 2*log(2)/3. (End)

Example

A 1 X 1 X 1 cube requires 12 rods.

Maple

A059986:=n->3*n*(n+1)^2; seq(A059986(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Mathematica

Table[EdgeCount[GridGraph[{n, n, n}]], {n, 39}] (* Geoffrey Critzer, May 17 2009 *)
Table[3 n (n + 1)^2, {n, 0, 50}] (* Wesley Ivan Hurt, May 13 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 12, 54, 144}, 20] (* Eric W. Weisstein, Mar 09 2024 *)
CoefficientList[Series[6 x (2 + x)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Mar 09 2024 *)

Prog

(Magma) [3*n*(n+1)^2: n in [0..50]]; // Wesley Ivan Hurt, May 13 2014
(PARI) a(n) = 3*n*(n+1)^2 \\ Charles R Greathouse IV, May 14 2014

Crossrefs

Cf. A045945, A117227.

Sequence in context: A022704 A372025 A060785 * A088941 A019582 A025204

Adjacent sequences: A059983 A059984 A059985 * A059987 A059988 A059989

Keyword

nonn,easy

Author

Laura Twomey (sxe15(AT)hotmail.com), Mar 07 2001

Extensions

More terms from Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005

Status

approved