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
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
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
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