A115099 a(0)=4, a(n) = 3*a(n-1) - 4.

4, 8, 20, 56, 164, 488, 1460, 4376, 13124, 39368, 118100, 354296, 1062884, 3188648, 9565940, 28697816, 86093444, 258280328, 774840980, 2324522936, 6973568804, 20920706408, 62762119220, 188286357656, 564859072964, 1694577218888, 5083731656660, 15251194969976

Offset: 0

Author

Miklos Kristof, Mar 02 2006

Keywords

• easy
• nonn

Comments

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Programs

Magma

[2*3^n+2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

Maple

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

Mathematica

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

PARI

Vec(4*(1-2*x)/((1-x)*(1-3*x)) + O(x^30)) \\ Colin Barker, May 31 2016

Formula

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

From Colin Barker, May 31 2016: (Start)

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

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

(End)

E.g.f.: 2*(1 + exp(2*x))*exp(x). - Ilya Gutkovskiy, May 31 2016

a(n) = 4 * A007051(n). - Alois P. Heinz, Jun 26 2023