A103453 a(n) = 0^n + 3^n - 1.

1, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442

Offset

0,2

Comments

A transform of 3^n under the matrix A103452.

a(n) is the number of moves required to solve a Towers of Hanoi puzzle of 3 towers in a line (no direct connection between the two towers on the ends) with n pieces to be moved from one end tower to the other. This is easily proved through demonstration. - Roderick Kimball, Nov 22 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..190

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Formula

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

a(n) = Sum_{k=0..n} A103452(n, k)*3^k.

a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*3^k.

From G. C. Greubel, Jun 18 2021: (Start)

E.g.f.: 1 - exp(x) + exp(3*x).

a(n) = [n=0] + 2*A003462(n). (End)

Mathematica

Table[If[n==0, 1, 3^n -1], {n, 0, 30}] (* G. C. Greubel, Jun 18 2021 *)
LinearRecurrence[{4, -3}, {1, 2, 8}, 30] (* Harvey P. Dale, Feb 13 2022 *)

Prog

(Magma) [0^n+3^n-1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011
(PARI) a(n) = if(n==0, 1, 3^n-1); \\ Altug Alkan, Nov 22 2015
(Sage) [3^n -1 +0^n for n in (0..30)] # G. C. Greubel, Jun 18 2021

Crossrefs

Cf. A103452.

Essentially identical to A024023.

Sequence in context: A224289 A124721 A279735 * A024023 A295137 A126966

Adjacent sequences: A103450 A103451 A103452 * A103454 A103455 A103456

Keyword

easy,nonn

Author

Paul Barry, Feb 06 2005

Status

approved