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A005666 Tower of Hanoi with 3 pegs and cyclic moves only (counterclockwise).
(Formerly M1755)
2
0, 2, 7, 21, 59, 163, 447, 1223, 3343, 9135, 24959, 68191, 186303, 508991, 1390591, 3799167, 10379519, 28357375, 77473791, 211662335, 578272255, 1579869183, 4316282879, 11792304127, 32217174015, 88018956287, 240472260607, 656982433791, 1794909388799 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 18.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..28.

J.-P. Allouche, Note on the cyclic towers of Hanoi, Theoret. Comput. Sci., 123 (1994), 3-7.

M. D. Atkinson, The Cyclic Towers of Hanoi, Info. Proc. Letters, 13 (1981), 118-119.

R. K. Guy, Letter to N. J. A. Sloane, 1976

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

D. G. Poole, The towers and triangles of Professor Claus (or, Pascal knows Hanoi), Math. Mag., 67 (1994), 323-344.

FORMULA

a(n) = (1/(4*s3))*((1+s3)^(n+2)-(1-s3)^(n+2))-1 where s3 = sqrt(3).

a(n) = A028859(n) - 1.

From Paul Zimmermann, Feb 07 2018: (Start)

a(n) = 2*a(n-1)+2*a(n-2)+3 (same recurrence as A005665).

a(n) = 2*a(n-1)+c(n-1)+2 where c(n) = 2*a(n-1)+1 stands for A005665.

(End)

MAPLE

A005666:=z*(2+z)/(z-1)/(2*z**2+2*z-1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

MATHEMATICA

CoefficientList[Series[z (2 + z)/(z - 1)/(2 z^2 + 2 z - 1), {z, 0, 22}], z] (* Michael De Vlieger, Sep 02 2015, translated from the Maple program *)

PROG

(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

CROSSREFS

Cf. A005665.

Sequence in context: A202027 A018036 A007050 * A291411 A159972 A106271

Adjacent sequences:  A005663 A005664 A005665 * A005667 A005668 A005669

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vincenzo Librandi, Sep 03 2015

Name clarified by Paul Zimmermann, Feb 09 2018

STATUS

approved

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Last modified October 21 13:05 EDT 2018. Contains 316422 sequences. (Running on oeis4.)