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A193248 Truncated dodecahedron, and truncated icosahedron with faces of centered polygons. 2
1, 93, 455, 1267, 2709, 4961, 8203, 12615, 18377, 25669, 34671, 45563, 58525, 73737, 91379, 111631, 134673, 160685, 189847, 222339, 258341, 298033, 341595, 389207, 441049, 497301, 558143, 623755, 694317, 770009, 851011, 937503, 1029665, 1127677, 1231719 (list; graph; refs; listen; history; text; internal format)
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

OEIS, (Centered_polygons) pyramidal numbers

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

Sequence in context: A237604 A045235 A116240 * A146090 A160174 A238693

Adjacent sequences:  A193245 A193246 A193247 * A193249 A193250 A193251

KEYWORD

nonn

AUTHOR

Craig Ferguson, Jul 19 2011

STATUS

approved

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Last modified May 25 00:59 EDT 2019. Contains 323534 sequences. (Running on oeis4.)