A033129 Base-2 digits are, in order, the first n terms of the periodic sequence with initial period [1,1,0].

0, 1, 3, 6, 13, 27, 54, 109, 219, 438, 877, 1755, 3510, 7021, 14043, 28086, 56173, 112347, 224694, 449389, 898779, 1797558, 3595117, 7190235, 14380470, 28760941, 57521883, 115043766, 230087533, 460175067, 920350134, 1840700269

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Number of moves to separate a Hanoi Tower into two towers of even resp. odd stones. - Martin von Gagern, May 26 2004

From Reinhard Zumkeller, Feb 22 2010: (Start)

Terms of A173593 with initial digits '11' in binary representation: a(n) = A173593(2*n-3) for n>0;

for n>0: a(3*n-1) = A083713(n);

a(n+1) - a(n) = abs(A078043(n)). (End)

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Mohammad Sajjad Hossain, reArrange.
• James Metz, Twists on the Tower of Hanoi, Math. Teacher, Vol. 107, No. 9 (2014), 712-715.
• Index entries for sequences related to Towers of Hanoi
• Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Crossrefs

Cf. A011655 (repeat 0,1,1), A289006 (the same in octal).

Cf. A057744, A294627 (first differences).

Programs

Mathematica

Table[(1/14)*(-9 - 2*(-1)^Floor[(2 n)/3] + (-1)^(1 + Floor[(1/3)*(7 + 2 n)]) + 3*2^(2 + n)), {n, 0, 100}] (* John M. Campbell, Dec 26 2016 *)
Table[FromDigits[PadRight[{},n,{1,1,0}],2],{n,0,40}] (* Harvey P. Dale, Oct 02 2022 *)

PARI

A033129(n)=3<<(n+1)\7 \\ M. F. Hasler, Jun 23 2017

Python

print([(6*2**n//7) for n in range(50)]) # Karl V. Keller, Jr., Jul 11 2022

Formula

From Paul Barry, Jan 23 2004: (Start)

Partial sums of abs(A078043).

G.f.: x*(1+x)/((1-x)*(1-2*x)*(1+x+x^2)) = x*(1+x)/(1-2*x-x^3+2*x^4).

a(n) = (6/7)*2^n - (4/21)*cos(2*Pi*n/3) - (2/21)*sqrt(3)*sin(2*Pi*n/3) - 2/3. (End)

a(n) = a(n-3) + 3 * 2^(n-3). - Martin von Gagern, May 26 2004

a(n+1) = 2*a(n) + 1 - 0^((a(n)+1) mod 4). - Reinhard Zumkeller, Feb 22 2010

a(n) = floor(2^(n+1)*3/7). - Jean-Marie Madiot, Oct 05 2012

a(n) = (1/14)*(-9 - 2*(-1)^floor((2n)/3) + (-1)^(floor((2*n + 7)/3) + 1) + 3*2^(n + 2)). - John M. Campbell, Dec 26 2016