1,2

Consider a figurate cubic number of the form a(n)=n^3. n is interpreted as the number of dots or nodes in each edge of the cube. Refer this cube as "central cube". Suppose one identical cube is attached to each of its six faces of the central cube. The resulting geometric structure consists of a total of seven arranged cubes so that each of the six surrounding cubes shares an entire face with the central cube. The overlapping dots along these shared faces are counted once. The number of dots in this configuration is given by the formula: a(n) = 7*n^3-6*n^2 for n>=1.

Jejemae S. Maque, "Augmented Cubic Numbers," Undergraduate Thesis, Bukidnon State University, 2024.

Noel B. Lacpao, Table of n, a(n) for n = 1..1000

Noel B. Lacpao, Illustration of the augmented cubic number structure for n=2

Noel B. Lacpao, Illustration of the augmented cubic number structure for n=3

Noel B. Lacpao, Colab notebook for generating augmented cubic numbers

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

a(n) = 7*n^3 - 6*n^2.

G.f.: x*(1 + 28*x + 13*x^2) / (1-x)^4.

For n=2, a(2) = 7*(2^3) - 6*(2^2) = 32.

For n=5, a(5) = 7*(5^3) - 6*(5^2) = 725.

seq(7*n^3 - 6*n^2, n=1..20);

Table[7 n^3 - 6 n^2, {n, 1, 20}]

(Python)

[7*n**3 - 6*n**2 for n in range(1, 21)]

Sequence in context: A044364 A044745 A297641 * A304380 A297887 A298492

Adjacent sequences: A382248 A382249 A382250 * A382252 A382253 A382254

nonn,easy

Noel B. Lacpao, May 17 2025

approved