OFFSET
1,2
COMMENTS
a(n-1) is the number of points in a cubic lattice of n^3 equally spaced points from which all the 12*n-16 points on the 12 edges are removed. - Luciano Ancora, Jun 25 2015
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 6*(n-1)^2 + (n-1)^3.
G.f.: x^2*(7+4*x-5*x^2)/(1-x)^4. - Colin Barker, Jul 29 2012
a(n-1) = n^3 - (12*n-16). - Luciano Ancora, Jun 25 2015
EXAMPLE
On a cube made of 3 X 3 X 3 smaller cubes, each of the 6 sides has 4 intersections of four edges and in the center, there are 8 intersections of six edges. 6 * 4 + 8 = 32, which is a(3).
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {0, 7, 32, 81}, 50] (* Vincenzo Librandi, Jun 27 2015 *)
PROG
(PARI) concat(0, Vec(x^2*(7+4*x-5*x^2)/(1-x)^4 + O(x^50))) \\ Michel Marcus, Jun 26 2015
(Magma) [6*(n-1)^2 + (n-1)^3: n in [1..40]]; // Vincenzo Librandi, Jun 27 2015
(Python) def a(n): return (n**3+3*n**2-9*n+5) # Torlach Rush, May 01 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 12 2007
STATUS
approved