A100774 a(n) = 2*(3^n - 1).

0, 4, 16, 52, 160, 484, 1456, 4372, 13120, 39364, 118096, 354292, 1062880, 3188644, 9565936, 28697812, 86093440, 258280324, 774840976, 2324522932, 6973568800, 20920706404, 62762119216, 188286357652, 564859072960, 1694577218884

Offset: 0

Author

Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005

Keywords

• easy
• nonn

Comments

a(n) is the number of steps which are made when generating all n-step nonreversing random walks that begin in a fixed point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice.

These are also the first local maxima reached in the Collatz trajectories of 2^n - 1. - David Rabahy, Oct 30 2017

Also the graph diameter of the n-Sierpinski carpet graph. - Eric W. Weisstein, Mar 13 2018

a(n) is the number of edge covers of F_{n,2}, which has adjacent vertices u and w, and n vertices each adjacent to both u and w. An edge cover is a subset of the edges where each vertex is adjacent to at least one vertex. To cover each of the n vertices v_i, we need to have at least the edge uv_i or wv_i or both, giving us three choices for each. We can then add the edge uw or not, which is 2*3^n choices. But we need to remove the case where all uv_i's were chosen and uw not chosen, and all ww_i's were chosen and uw not chosen. - Feryal Alayont, Jun 17 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..196
• Eric Weisstein's World of Mathematics, Graph Diameter
• Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A003462, A007051, A024023, A029858, A034472, A048473, A058481, A067771, A079004, A115099, A134931.

Programs

Magma

[2*(3^n - 1): n in [0..25] ]; // Vincenzo Librandi, Apr 30 2011

Mathematica

Table[2 (3^n - 1), {n, 0, 24}] (* Alonso del Arte, Nov 08 2012 *)
2 (3^Range[0, 20] - 1) (* Eric W. Weisstein, Mar 13 2018 *)
LinearRecurrence[{4, -3}, {4, 16}, {0, 20}] (* Eric W. Weisstein, Mar 13 2018 *)
CoefficientList[Series[4 x/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 13 2018 *)

Maxima

A100774(n):=2*(3^n - 1)$
makelist(A100774(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

PARI

a(n)=2*3^n-2 \\ Charles R Greathouse IV, Nov 09 2012

Formula

a(n) = 2*(3^n - 1).

a(n) = 4*Sum_{i=0..n-1} 3^i.

a(n) = 4*A003462(n).

a(n) = A048473(n) - 1. - Paul Curtz, Jan 19 2009

G.f.: 4*x/((1-x)*(1-3*x)). - Eric W. Weisstein, Mar 13 2018

a(n) = 4*a(n-1) - 3*a(n-2). - Eric W. Weisstein, Mar 13 2018

From Elmo R. Oliveira, Dec 06 2023: (Start)

a(n) = 2*A024023(n).

a(n) = 3*a(n-1) + 4 for n>0.

E.g.f.: 2*(exp(3*x) - exp(x)). (End)