A103453 - OEIS

%I #38 Jun 27 2023 11:10:46

%S 1,2,8,26,80,242,728,2186,6560,19682,59048,177146,531440,1594322,

%T 4782968,14348906,43046720,129140162,387420488,1162261466,3486784400,

%U 10460353202,31381059608,94143178826,282429536480,847288609442

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

%C A transform of 3^n under the matrix A103452.

%C 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

%H Vincenzo Librandi, <a href="/A103453/b103453.txt">Table of n, a(n) for n = 0..190</a>

%H Steven Schlicker, Roman Vasquez, and Rachel Wofford, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Wofford/wofford4.html">Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

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

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

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

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

%F From _G. C. Greubel_, Jun 18 2021: (Start)

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

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

%t Table[If[n==0, 1, 3^n -1], {n, 0, 30}] (* _G. C. Greubel_, Jun 18 2021 *)

%t LinearRecurrence[{4,-3},{1,2,8},30] (* _Harvey P. Dale_, Feb 13 2022 *)

%o (Magma) [0^n+3^n-1: n in [0..30] ]; // _Vincenzo Librandi_, Apr 30 2011

%o (PARI) a(n) = if(n==0, 1, 3^n-1); \\ _Altug Alkan_, Nov 22 2015

%o (Sage) [3^n -1 +0^n for n in (0..30)] # _G. C. Greubel_, Jun 18 2021

%Y Cf. A103452.

%Y Essentially identical to A024023.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 06 2005