

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.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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)^3n^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(n1)  3*a(n2) + a(n3) 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: A098136 A060793 A169823 * A177871 A252953 A309315
Adjacent sequences: A309839 A309840 A309841 * A309843 A309844 A309845


KEYWORD

nonn,easy


AUTHOR

Anthony D. Santamaria, Nov 18 2019


STATUS

approved



