OFFSET
0,2
COMMENTS
Original name was: Tower of Hanoi with 3 pegs and cyclic moves only (counterclockwise). - Jianing Song, Nov 01 2024
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
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
D. G. Poole, The towers and triangles of Professor Claus (or, Pascal knows Hanoi), Math. Mag., 67 (1994), 323-344.
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case (ii), cyclic.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).
FORMULA
a(n) = (1/(4*s3))*((1+s3)^(n+2)-(1-s3)^(n+2))-1 where s3 = sqrt(3).
a(n) = A028859(n) - 1.
G.f.: x*(2+x) / ( (x-1)*(2*x^2+2*x-1) ). - Simon Plouffe in his 1992 dissertation
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)
E.g.f.: exp(x)*(3*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 3)/3. - Stefano Spezia, Apr 11 2025
MATHEMATICA
CoefficientList[Series[z (2 + z)/(z - 1)/(2 z^2 + 2 z - 1), {z, 0, 22}], z] (* Michael De Vlieger, Sep 02 2015 *)
LinearRecurrence[{3, 0, -2}, {0, 2, 7}, 30] (* Harvey P. Dale, Jul 28 2025 *)
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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Name clarified by Paul Zimmermann, Feb 09 2018
STATUS
approved
