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a(0)=4, a(n) = 3*a(n-1) - 4.
11

%I #23 Jun 26 2023 19:32:32

%S 4,8,20,56,164,488,1460,4376,13124,39368,118100,354296,1062884,

%T 3188648,9565940,28697816,86093444,258280328,774840980,2324522936,

%U 6973568804,20920706408,62762119220,188286357656,564859072964,1694577218888,5083731656660,15251194969976

%N a(0)=4, a(n) = 3*a(n-1) - 4.

%C A tetrahedron has 4 faces. Cut every corner so that we get triangular faces; the resulting polyhedron has 8 faces. Repeating this procedure gives polyhedra with 4, 8, 20, 56, etc. faces.

%H Vincenzo Librandi, <a href="/A115099/b115099.txt">Table of n, a(n) for n = 0..300</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).

%F a(n) = 2*3^n + 2.

%F From _Colin Barker_, May 31 2016: (Start)

%F a(n) = 4*a(n-1)-3*a(n-2) for n>1.

%F G.f.: 4*(1-2*x) / ((1-x)*(1-3*x)).

%F (End)

%F E.g.f.: 2*(1 + exp(2*x))*exp(x). - _Ilya Gutkovskiy_, May 31 2016

%F a(n) = 4 * A007051(n). - _Alois P. Heinz_, Jun 26 2023

%p seq(2*3^i+2,i=0..30);

%t a=4;lst={a};Do[a=a*3-4;AppendTo[lst,a],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008 *)

%o (Magma) [2*3^n+2: n in [0..30]]; // _Vincenzo Librandi_, Jun 05 2011

%o (PARI) Vec(4*(1-2*x)/((1-x)*(1-3*x)) + O(x^30)) \\ _Colin Barker_, May 31 2016

%Y Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931. - _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008

%K easy,nonn

%O 0,1

%A _Miklos Kristof_, Mar 02 2006