OFFSET
1,2
COMMENTS
The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated dodecahedron or truncated icosahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448)and centered decagons (A062786)] & [centered hexagons (A003215) and centered pentagons (A005891)] respectively.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 30*n^3 - 45*n^2 + 17*n - 1.
G.f.: x*(1+x)*(x^2 + 88*x + 1) / (x-1)^4. - R. J. Mathar, Aug 26 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=93, a(3)=455, a(4)=1267. - Harvey P. Dale, Aug 28 2011
E.g.f.: 1 - (1 - 2*x - 45*x^2 - 30*x^3)*exp(x). - G. C. Greubel, Nov 10 2018
MATHEMATICA
Table[30n^3-45n^2+17n-1, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 93, 455, 1267}, 40] (* Harvey P. Dale, Aug 28 2011 *)
PROG
(Excel) copy and paste the formula =30 *ROW()^3-45 *ROW()^2+17*ROW()-1 fill down to desired size.
(Magma) [30*n^3-45*n^2+17*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011
(PARI) vector(40, n, 30*n^3 - 45*n^2 + 17*n - 1) \\ G. C. Greubel, Nov 10 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Craig Ferguson, Jul 19 2011
STATUS
approved