A126562 Number of intersections of at least four edges in a cube of n X n X n smaller cubes.

0, 7, 32, 81, 160, 275, 432, 637, 896, 1215, 1600, 2057, 2592, 3211, 3920, 4725, 5632, 6647, 7776, 9025, 10400, 11907, 13552, 15341, 17280, 19375, 21632, 24057, 26656, 29435, 32400, 35557, 38912, 42471, 46240, 50225, 54432, 58867, 63536, 68445

Offset: 1

Author

Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 12 2007

Keywords

• nonn
• easy

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

Examples

			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).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A017617.

Programs

Magma

[6*(n-1)^2 + (n-1)^3: n in [1..40]]; // Vincenzo Librandi, Jun 27 2015

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {0, 7, 32, 81}, 50] (* Vincenzo Librandi, Jun 27 2015 *)

PARI

concat(0, Vec(x^2*(7+4*x-5*x^2)/(1-x)^4 + O(x^50))) \\ Michel Marcus, Jun 26 2015

Python

def a(n): return (n**3+3*n**2-9*n+5) # Torlach Rush, May 01 2024

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