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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
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.
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
Sequence in context: A371037 A334761 A169823 * A177871 A377418 A334382
KEYWORD
nonn,easy
AUTHOR
STATUS
approved