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A059986 Number of rods required to make a 3-D cube of side length n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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

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) = A059986(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

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[M[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 *)

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.

Sequence in context: A000735 A022704 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

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Last modified October 23 18:50 EDT 2018. Contains 316530 sequences. (Running on oeis4.)