cp's OEIS Frontend

A005666 Minimal number of moves for the cyclic variant of the Towers of Hanoi for 3 pegs and n disks, with the final peg two steps away.

%I A005666 M1755 #81 Jul 28 2025 18:03:18
%S A005666 0,2,7,21,59,163,447,1223,3343,9135,24959,68191,186303,508991,1390591,
%T A005666 3799167,10379519,28357375,77473791,211662335,578272255,1579869183,
%U A005666 4316282879,11792304127,32217174015,88018956287,240472260607,656982433791,1794909388799
%N A005666 Minimal number of moves for the cyclic variant of the Towers of Hanoi for 3 pegs and n disks, with the final peg two steps away.
%C A005666 Original name was: Tower of Hanoi with 3 pegs and cyclic moves only (counterclockwise). - _Jianing Song_, Nov 01 2024
%D A005666 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 18.
%D A005666 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005666 J.-P. Allouche, <a href="http://dx.doi.org/10.1016/0304-3975(94)90064-7">Note on the cyclic towers of Hanoi</a>, Theoret. Comput. Sci., 123 (1994), 3-7.
%H A005666 M. D. Atkinson, <a href="http://www.cs.otago.ac.nz/staffpriv/mike/Papers/Hanoi/CyclicHanoi.pdf">The Cyclic Towers of Hanoi</a>, Info. Proc. Letters, 13 (1981), 118-119.
%H A005666 R. K. Guy, <a href="/A005665/a005665_1.pdf">Letter to N. J. A. Sloane, 1976</a>
%H A005666 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005666 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005666 D. G. Poole, <a href="http://www.jstor.org/stable/2690991">The towers and triangles of Professor Claus (or, Pascal knows Hanoi)</a>, Math. Mag., 67 (1994), 323-344.
%H A005666 Amir Sapir, <a href="https://doi.org/10.1093/comjnl/47.1.20">The Tower of Hanoi with Forbidden Moves</a>, The Computer J. 47 (1) (2004) 20, case (ii), cyclic.
%H A005666 <a href="/index/To#Hanoi">Index entries for sequences related to Towers of Hanoi</a>
%H A005666 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-2).
%F A005666 a(n) = (1/(4*s3))*((1+s3)^(n+2)-(1-s3)^(n+2))-1 where s3 = sqrt(3).
%F A005666 a(n) = A028859(n) - 1.
%F A005666 G.f.: x*(2+x) / ( (x-1)*(2*x^2+2*x-1) ). - _Simon Plouffe_ in his 1992 dissertation
%F A005666 From _Paul Zimmermann_, Feb 07 2018: (Start)
%F A005666 a(n) = 2*a(n-1)+2*a(n-2)+3 (same recurrence as A005665).
%F A005666 a(n) = 2*a(n-1)+c(n-1)+2 where c(n) = 2*a(n-1)+1 stands for A005665. (End)
%F A005666 E.g.f.: exp(x)*(3*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 3)/3. - _Stefano Spezia_, Apr 11 2025
%t A005666 CoefficientList[Series[z (2 + z)/(z - 1)/(2 z^2 + 2 z - 1), {z, 0, 22}], z] (* _Michael De Vlieger_, Sep 02 2015 *)
%t A005666 LinearRecurrence[{3,0,-2},{0,2,7},30] (* _Harvey P. Dale_, Jul 28 2025 *)
%o A005666 (Magma) [Floor((1/(4*Sqrt(3)))*((1+Sqrt(3))^(n+2)-(1-Sqrt(3))^(n+2))-1): n in [0..30]]; // _Vincenzo Librandi_, Sep 03 2015
%Y A005666 Cf. A005665, A052945 (first differences).
%Y A005666 Cf. A338024, A292764, A338089 (4 pegs).
%K A005666 nonn,nice,easy
%O A005666 0,2
%A A005666 _N. J. A. Sloane_
%E A005666 Name clarified by _Paul Zimmermann_, Feb 09 2018
%E A005666 New name based on the name of A338024, A292764, and A338089 by _Jianing Song_, Nov 01 2024