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 A193218 #59 Sep 08 2022 08:45:58
%S A193218 1,21,95,259,549,1001,1651,2535,3689,5149,6951,9131,11725,14769,18299,
%T A193218 22351,26961,32165,37999,44499,51701,59641,68355,77879,88249,99501,
%U A193218 111671,124795,138909,154049,170251,187551,205985,225589,246399,268451,291781,316425
%N A193218 Number of vertices in truncated tetrahedron with faces that are centered polygons.
%C A193218 The sequence starts with a central vertex and expands outward with (n-1) centered polygonal pyramids producing a truncated tetrahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon in each face. For centered triangles see A005448 and centered hexagons A003215.
%C A193218 This sequence is the 18th in the series (1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496 and t = 36. While adjusting for offsets, the beginning sequence A049480 is generated by adding the square pyramidal numbers A000330 to the odd numbers A005408 and each subsequent sequence is found by adding another set of square pyramidals A000330. (T/2) * A000330(n) + A005408(n). At 30 * A000330 + A005408 = centered dodecahedral numbers, 36 * A000330 + A005408 = A193228 truncated octahedron and 90 * A000330 + A005408 = A193248 = truncated icosahedron and dodecahedron. All five of the "Centered Platonic Solids" numbers sequences are in this series of sequences. Also 4 out of five of the "truncated" platonic solid number sequences are in this series. - _Bruce J. Nicholson_, Jul 06 2018
%C A193218 It would be good to have a detailed description of how the sequence is constructed. Maybe in the Examples section? - _N. J. A. Sloane_, Sep 07 2018
%H A193218 Vincenzo Librandi, <a href="/A193218/b193218.txt">Table of n, a(n) for n = 1..10000</a>
%H A193218 OEIS, <a href="http://oeis.org/wiki/(Centered_polygons)_pyramidal_numbers"> (Centered_polygons) pyramidal numbers</a>
%H A193218 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>
%H A193218 Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>
%H A193218 Wikipedia, <a href="http://en.wikipedia.org/wiki/Centered_polygonal_number">Centered polygonal number</a>
%H A193218 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A193218 a(n) = 6*n^3 - 9*n^2 + 5*n - 1.
%F A193218 G.f.: x*(1+x)*(x^2+16*x+1) / (1-x)^4. - _R. J. Mathar_, Aug 26 2011
%F A193218 a(n) = 18 * A000330(n-1) + A005408(n-1) = A063496(n) + A006331(n-1). - _Bruce J. Nicholson_, Jul 06 2018
%t A193218 Table[6 n^3 - 9 n^2 + 5 n - 1, {n, 35}] (* _Alonso del Arte_, Jul 18 2011 *)
%t A193218 CoefficientList[Series[(1+x)*(x^2+16*x+1)/(1-x)^4, {x, 0, 50}], x] (* _Stefano Spezia_, Sep 04 2018 *)
%o A193218 (Magma) [6*n^3-9*n^2+5*n-1: n in [1..40]]; // _Vincenzo Librandi_, Aug 30 2011
%Y A193218 Cf. A260810 (partial sums).
%K A193218 nonn,easy
%O A193218 1,2
%A A193218 _Craig Ferguson_, Jul 18 2011