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%I #26 Aug 29 2021 02:03:55
%S 3,72,1419,26580,487839,8867088,160391235,2894149932,52158948999,
%T 939440707560,16915155908523,304519845578052,5481780715831215,
%U 98675865000853056,1776199882077971859,31971906699808312284,575497100061532320855,10358972816581956751128
%N Wiener index of the n-Hanoi graph.
%C Sequence gives 1/2 of the total number of moves summed over all starting and finishing positions in the tower of Hanoi puzzle with n disks. For just the total number of moves from all starting positions to the standard finish position see A060589.
%H Andrew Howroyd, <a href="/A290004/b290004.txt">Table of n, a(n) for n = 1..100</a>
%H T. Chan, <a href="http://dx.doi.org/10.1080/00207168908803728">A statistical analysis of the towers of Hanoi problem</a>, Internat. J. Comput. Math. 28: 57-65.
%H A. Hinz, <a href="https://doi.org/10.5169/seals-57378">The Tower of Hanoi</a>, L'Enseignement Mathématique, 35: 289-321.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tower_of_Hanoi">Tower of Hanoi</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (35, -395, 1761, -2916, 972).
%H <a href="/index/To#Hanoi">Index entries for sequences related to Towers of Hanoi</a>
%F a(n) = 35*a(n-1) - 395*a(n-2) + 1761*a(n-3) - 2916*a(n-4) + 972*a(n-5).
%F G.f.: 3*x*(1 - 11*x + 28*x^2 + 24*x^3)/((1 - 3*x)*(1 - 9*x)*(1 - 18*x)*(1 - 5*x + 2*x^2)). [Corrected by _Georg Fischer_, May 19 2019]
%t (* Start from _Eric W. Weisstein_, Sep 07 2017 *)
%t Table[3/1003 2^-n ((34 - 3 Sqrt[17]) (5 - Sqrt[17])^n + (5 + Sqrt[17])^n (34 + 3 Sqrt[17])) - 3^(n + 1)/10 + 9^n (233/885 2^n - 1/6), {n, 10}]
%t LinearRecurrence[{35, -395, 1761, -2916, 972}, {3, 72, 1419, 26580, 487839}, 20]
%t CoefficientList[Series[(3 (1 - 11 x + 28 x^2 + 24 x^3))/((1 - 3 x) (1 - 9 x) (1 - 18 x) (1 - 5 x + 2 x^2)), {x, 0, 20}], x]
%t (* End *)
%o (PARI) Vec(3*(1 - 11*x + 28*x^2 + 24*x^3)/((1 - 3*x)*(1 - 9*x)*(1 - 18*x)*(1 - 5*x + 2*x^2)) + O(x^20))
%Y Cf. A060589, A290129.
%K nonn,easy
%O 1,1
%A _Andrew Howroyd_, Sep 03 2017