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

Author

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

Keywords

• nonn
• easy

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

Examples

			A 1 X 1 X 1 cube requires 12 rods.
		

Links

• 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).

Crossrefs

Cf. A045945, A117227.

Programs

Magma

[3*n*(n+1)^2: n in [0..50]]; // Wesley Ivan Hurt, May 13 2014

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 *)

PARI

a(n) = 3*n*(n+1)^2 \\ Charles R Greathouse IV, May 14 2014

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)

Extensions

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