login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Base-2 digits are, in order, the first n terms of the periodic sequence with initial period [1,1,0].
9

%I #63 Oct 02 2022 18:20:51

%S 0,1,3,6,13,27,54,109,219,438,877,1755,3510,7021,14043,28086,56173,

%T 112347,224694,449389,898779,1797558,3595117,7190235,14380470,

%U 28760941,57521883,115043766,230087533,460175067,920350134,1840700269

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

%C Number of moves to separate a Hanoi Tower into two towers of even resp. odd stones. - _Martin von Gagern_, May 26 2004

%C From _Reinhard Zumkeller_, Feb 22 2010: (Start)

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

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

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

%H Harvey P. Dale, <a href="/A033129/b033129.txt">Table of n, a(n) for n = 0..1000</a>

%H Mohammad Sajjad Hossain, <a href="https://web.archive.org/web/20050827075933/http://online-judge.uva.es/p/v106/10658.html">reArrange</a>.

%H James Metz, <a href="http://www.jstor.org/stable/10.5951/mathteacher.107.9.0712">Twists on the Tower of Hanoi</a>, Math. Teacher, Vol. 107, No. 9 (2014), 712-715.

%H <a href="/index/To#Hanoi">Index entries for sequences related to Towers of Hanoi</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,-2).

%F From _Paul Barry_, Jan 23 2004: (Start)

%F Partial sums of abs(A078043).

%F 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).

%F 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)

%F a(n) = a(n-3) + 3 * 2^(n-3). - _Martin von Gagern_, May 26 2004

%F a(n+1) = 2*a(n) + 1 - 0^((a(n)+1) mod 4). - _Reinhard Zumkeller_, Feb 22 2010

%F a(n) = floor(2^(n+1)*3/7). - _Jean-Marie Madiot_, Oct 05 2012

%F 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

%t 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 *)

%t Table[FromDigits[PadRight[{},n,{1,1,0}],2],{n,0,40}] (* _Harvey P. Dale_, Oct 02 2022 *)

%o (PARI) A033129(n)=3<<(n+1)\7 \\ _M. F. Hasler_, Jun 23 2017

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

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

%Y Cf. A057744, A294627 (first differences).

%K nonn,base,easy

%O 0,3

%A _Clark Kimberling_