A033120 Base-2 digits of a(n) are, in order, the first n terms of the periodic sequence with initial period 1,0,1.

1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811, 93622, 187245, 374491, 748982, 1497965, 2995931, 5991862, 11983725, 23967451, 47934902, 95869805, 191739611, 383479222, 766958445, 1533916891

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Minimal number of moves required, under the proviso of a classical tower-of-Hanoi game, to segregate an initial n-disc peg into even and odd numbered discs pegs. - Lekraj Beedassy, Sep 12 2006

References

• B. Averbach & O. Chein, "A Variant Of The Tower Of Brahma" in 'The Journal of Recreational Mathematics', pp. 48-55, vol. 33, no. 1, 2004-5, Baywood, NY.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Crossrefs

Cf. A023001, A033137 (similar in base 10).

Programs

Mathematica

Table[FromDigits[PadRight[{},n,{1,0,1}],2],{n,40}] (* Harvey P. Dale, Aug 26 2016 *)

PARI

a(n)=if(n%3==0,5*8^(n/3)-5,if(n%3==1,10*8^((n-1)/3)-3,20*8^((n-2)/3)-6))/7 \\ Ralf Stephan

PARI

a(n)=(5*2^n)\7 \\ Tani Akinari, Jul 15 2014

Formula

From Ralf Stephan, May 05 2004: (Start)

a(3*n) = (5*8^n - 5)/7, a(3*n+1) = (10*8^n - 3)/7, a(3*n+2) = (20*8^n - 6)/7.

G.f.: (1+x^2)/((1-x)*(1-2*x)*(1+x+x^2)). (End)

a(n) = a(n-6) + 45*2^(n-6). - Lekraj Beedassy, Sep 12 2006

The following recurrence produces this sequence: if(n==1) a(n)=1; else if(n%3==2) a(n)=a(n-1)*2; otherwise a(n)=a(n-1)*2+1. - Piotr Kakol, Jan 24 2011 (in an email message to N. J. A. Sloane).

a(n) = floor( (5/7)*2^n ). - Tani Akinari, Jul 15 2014

From Jorijn Lamberink and Paul van de Veen, Oct 14 2019: (Start)

a(n) = T(n-1) + 1 + T(n-3) + 1 + a(n-3), where T(n) = A000225(n) = 2^n-1 is the number of moves for a classic Tower of Hanoi with n discs.

a(n) = (5/8)*2^n + a(n-3).

a(n) = (5/7)*2^n - 2/3 - (1/21)*cos((2/3)*Pi*n) + (1/7)*sqrt(3)*sin((2/3)*Pi*n). (End)

Extensions

More terms from Lekraj Beedassy, Sep 12 2006