A382251 a(n) = 7*n^3 - 6*n^2.

1, 32, 135, 352, 725, 1296, 2107, 3200, 4617, 6400, 8591, 11232, 14365, 18032, 22275, 27136, 32657, 38880, 45847, 53600, 62181, 71632, 81995, 93312, 105625, 118976, 133407, 148960, 165677, 183600, 202771, 223232, 245025, 268192, 292775, 318816, 346357, 375440, 406107, 438400, 472361

Offset: 1

Author

Noel B. Lacpao, May 17 2025

Keywords

• nonn
• easy

Comments

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.

Examples

			For n=2, a(2) = 7*(2^3) - 6*(2^2) = 32.
For n=5, a(5) = 7*(5^3) - 6*(5^2) = 725.
		

References

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

Links

• 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).

Programs

Maple

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

Mathematica

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

Python

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

Formula

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

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