%I #34 Sep 08 2022 08:45:58

%S 1,39,185,511,1089,1991,3289,5055,7361,10279,13881,18239,23425,29511,

%T 36569,44671,53889,64295,75961,88959,103361,119239,136665,155711,

%U 176449,198951,223289,249535,277761,308039,340441,375039,411905,451111,492729,536831,583489

%N Truncated octahedron with faces of centered polygons.

%C The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated octahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered squares (A001844) and centered hexagons (A003215)]

%H Vincenzo Librandi, <a href="/A193228/b193228.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS, <a href="http://oeis.org/wiki/(Centered_polygons)_pyramidal_numbers"> (Centered_polygons) pyramidal numbers</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Centered_polygonal_number">Centered polygonal number</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 12*n^3 - 18*n^2 + 8*n - 1.

%F G.f.: x*(1+x)*(x^2 + 34*x + 1) / (x-1)^4. - _R. J. Mathar_, Aug 26 2011

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=39, a(2)=185, a(3)=511. - _Harvey P. Dale_, Aug 27 2011

%F E.g.f.: 1 - (1 - 2*x - 18*x^2 - 12*x^3)*exp(x). - _G. C. Greubel_, Nov 10 2018

%t Table[12n^3-18n^2+8n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,39,185,511},40] (* _Harvey P. Dale_, Aug 27 2011 *)

%o (Excel) (copy and paste the following formula =12*ROW()^3-18*ROW()^2+8*ROW()-1 fill down to desired size.)

%o (Magma) [12*n^3-18*n^2+8*n-1: n in [1..50]]; // _Vincenzo Librandi_, Aug 30 2011

%o (PARI) vector(40, n, 12*n^3 - 18*n^2 + 8*n - 1) \\ _G. C. Greubel_, Nov 10 2018

%K nonn,easy

%O 1,2

%A _Craig Ferguson_, Jul 18 2011