OFFSET
0,3
COMMENTS
Original name was: Tower of Hanoi with 3 pegs and cyclic moves only (clockwise). - Jianing Song, Nov 01 2024
This looks like sequence A(0,1;2,2;3) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
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
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Alejandre, Legend of Towers of Hanoi
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
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 249. Book's website
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
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.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).
FORMULA
G.f.: x*(1+2*x)/((1-x)*(1-2*x-2*x^2)). - Simon Plouffe in his 1992 dissertation
From Paul Barry, Sep 05 2006: (Start)
a(n) = ((sqrt(3)+1)^(n+1) + (sqrt(3)-1)^(n+1)*(-1)^n)*sqrt(3)/6 - 1. (End)
a(n) = 2*a(n-1) + 2*a(n-2) + 3. - John W. Layman
a(n) = (1/(2*s3))*((1+s3)^(n+1) - (1-s3)^(n+1)) - 1 where s3 = sqrt(3).
a(n) = 3*a(n-1) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5 (from the given o.g.f.). Observed by Gary Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(n) = 2*A005666(n-1) + 1. - Michel Marcus, Nov 02 2012
a(n) = Sum_{k=1..n} A026150(k). - Ivan N. Ianakiev, Nov 22 2019
E.g.f.: (1/3)*exp(x)*(-3 + 3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Nov 22 2019
MATHEMATICA
a[n_] := Simplify[ ((1 + Sqrt[3])^(n+1) - (1 - Sqrt[3])^(n+1))*Sqrt[3]/6 - 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 14 2011, after Paul Barry *)
LinearRecurrence[{3, 0, -2}, {0, 1, 5}, 40] (* Harvey P. Dale, Mar 30 2015 *)
PROG
(Magma) [Floor(((Sqrt(3)+1)^(n+1)+(Sqrt(3)-1)^(n+1)*(-1)^n)*Sqrt(3)/6-1): n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011
(Haskell)
a005665 n = a005665_list !! (n-1)
a005665_list = 0 : 1 : 5 : zipWith (-)
(map (* 3) $ drop 2 a005665_list) (map (* 2) a005665_list)
-- Reinhard Zumkeller, May 01 2013
(PARI) a(n)=([0, 1, 0; 0, 0, 1; -2, 0, 3]^n*[0; 1; 5])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vincenzo Librandi, Aug 19 2011
Name clarified by Paul Zimmermann, Feb 21 2018
STATUS
approved