This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382251 #64 Jun 15 2025 17:58:55
%S A382251 1,32,135,352,725,1296,2107,3200,4617,6400,8591,11232,14365,18032,
%T A382251 22275,27136,32657,38880,45847,53600,62181,71632,81995,93312,105625,
%U A382251 118976,133407,148960,165677,183600,202771,223232,245025,268192,292775,318816,346357,375440,406107,438400,472361
%N A382251 a(n) = 7*n^3 - 6*n^2.
%C A382251 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.
%D A382251 Jejemae S. Maque, "Augmented Cubic Numbers," Undergraduate Thesis, Bukidnon State University, 2024.
%H A382251 Noel B. Lacpao, <a href="/A382251/b382251.txt">Table of n, a(n) for n = 1..1000</a>
%H A382251 Noel B. Lacpao, <a href="/A382251/a382251_7.png">Illustration of the augmented cubic number structure for n=2</a>
%H A382251 Noel B. Lacpao, <a href="/A382251/a382251_10.png">Illustration of the augmented cubic number structure for n=3</a>
%H A382251 Noel B. Lacpao, <a href="https://colab.research.google.com/drive/1gcHVr_Qa67s3dBNvXwrRMRApchtm1PEt?usp=sharing">Colab notebook for generating augmented cubic numbers</a>
%H A382251 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A382251 a(n) = 7*n^3 - 6*n^2.
%F A382251 G.f.: x*(1 + 28*x + 13*x^2) / (1-x)^4.
%e A382251 For n=2, a(2) = 7*(2^3) - 6*(2^2) = 32.
%e A382251 For n=5, a(5) = 7*(5^3) - 6*(5^2) = 725.
%p A382251 seq(7*n^3 - 6*n^2, n=1..20);
%t A382251 Table[7 n^3 - 6 n^2, {n, 1, 20}]
%o A382251 (Python)
%o A382251 [7*n**3 - 6*n**2 for n in range(1, 21)]
%K A382251 nonn,easy
%O A382251 1,2
%A A382251 _Noel B. Lacpao_, May 17 2025