A309842 a(n) is the total surface area of a hollow cubic block (defined as a block with a shell thickness of 1 cube) where n is the edge length of the removed volume.

60, 120, 204, 312, 444, 600, 780, 984, 1212, 1464, 1740, 2040, 2364, 2712, 3084, 3480, 3900, 4344, 4812, 5304, 5820, 6360, 6924, 7512, 8124, 8760, 9420, 10104, 10812, 11544, 12300, 13080, 13884, 14712, 15564, 16440, 17340, 18264, 19212, 20184, 21180, 22200, 23244

Offset

1,1

Comments

A cubic block is made of (n+2)^3 unit cubes, beginning with n=1. A hollow cubic block is a cubic block with all internal cubes removed (making its thickness 1), so its volume is (n+2)^3-n^3. The external edge length of a hollow cube is n+2, while the internal hollow volume edge length is n. The external surface area is 6*(n+2)^2 while the internal surface area is 6*n^2, therefore the total surface area is 6*(n+2)^2+6*n^2.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 12*n^2 + 24*n + 24.

a(n) = A033581(n) + A033581(n+2).

a(n) = 12*A002522(n+1).

From Colin Barker, Nov 24 2019: (Start)

G.f.: 12*x*(5 - 5*x + 2*x^2) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.

(End)

Example

The minimum cubic block that can be hollow is of volume 27, it is made hollow by removing the center cube. The external surface area is 3*3*6=54, the internal surface area is 1*1*6=6 so the total area is 60. a(1) = 60.

Prog

(PARI) Vec(12*x*(5 - 5*x + 2*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Nov 24 2019

Crossrefs

Cf. A002522, A033581.

Sequence in context: A371037 A334761 A169823 * A177871 A377418 A334382

Adjacent sequences: A309839 A309840 A309841 * A309843 A309844 A309845

Keyword

nonn,easy

Author

Anthony D. Santamaria, Nov 18 2019

Status

approved