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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

%I

%S 60,120,204,312,444,600,780,984,1212,1464,1740,2040,2364,2712,3084,

%T 3480,3900,4344,4812,5304,5820,6360,6924,7512,8124,8760,9420,10104,

%U 10812,11544,12300,13080,13884,14712,15564,16440,17340,18264,19212,20184,21180,22200,23244

%N 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.

%C 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.

%H Colin Barker, <a href="/A309842/b309842.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

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

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

%F From _Colin Barker_, Nov 24 2019: (Start)

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

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

%F (End)

%e 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.

%o (PARI) Vec(12*x*(5 - 5*x + 2*x^2) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Nov 24 2019

%Y Cf. A002522, A033581.

%K nonn,easy

%O 1,1

%A _Anthony D. Santamaria_, Nov 18 2019

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Last modified April 7 00:55 EDT 2020. Contains 333291 sequences. (Running on oeis4.)