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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 (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)^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: 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

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Last modified February 26 21:58 EST 2020. Contains 332295 sequences. (Running on oeis4.)