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A059986
Number of rods required to make a 3-D cube of side length n.
4
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
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Intersection Number.
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
Sequence in context: A022704 A372025 A060785 * A088941 A019582 A025204
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