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) = 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
KEYWORD
nonn,easy
AUTHOR
Anthony D. Santamaria, Nov 18 2019
STATUS
approved