The binary carry sequence gives the exponent of the
highest power of 2 dividing a positive
integer,
equivalently the number of trailing 0s in its binary representation.
For ,
2, ..., the first few are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ... (OEIS A007814).
This is one less than the label of the disk moved at the th step in the standard optimal three-rod solution to the tower of Hanoi problem: 1, 2, 1, 3, 1, 2, 1, 4, 1,
2, 1, ... (OEIS A001511). The parity
of this disk-label sequence is given by 1, 0, 1, 1,
1, 0, 1, 0, 1, 0, 1, 1, ... (OEIS A035263)
which, amazingly, also corresponds to the accumulation
point of
cycles through successive bifurcations.
Atanassov, K. "On the 37th and the 38th Smarandache Problems." Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria5,
83-85, 1999.Atanassov, K. On
Some of the Smarandache's Problems. Lupton, AZ: American Research Press,
pp. 16-21, 1999.Derrida, B.; Gervois, A.; and Pomeau, Y. "Iteration
of Endomorphisms on the Real Axis and Representation of Number." Ann. Inst.
Henri Poincaré, Section A: Physique Théorique29, 305-356,
1978.Karamanos, K. and Nicolis, G. "Symbolic Dynamics and Entropy
Analysis of Feigenbaum Limit Sets." Chaos, Solitons, Fractals10,
1135-1150, 1999.Metropolis, M.; Stein, M. L.; and Stein, P. R.
"On Finite Limit Sets for Transformations on the Unit Interval." J.
Combin. Th. A15, 25-44, 1973.Sloane, N. J. A.
Sequences A001511/M0127, A007814,
and A035263 in "The On-Line Encyclopedia
of Integer Sequences."Smarandache, F. Only
Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993.Vitanyi,
P. M. B. "An Optimal Simulation of Counter Machines." SIAM
J. Comput.14, 1-33, 1985.
gives the exponent of the
highest power of 2 dividing a positive
integer 
,
equivalently the number of trailing 0s in its binary representation.
For 
,
2, ..., the first few are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ... (OEIS A007814).
th step in the standard optimal three-rod solution to the tower of Hanoi problem: 1, 2, 1, 3, 1, 2, 1, 4, 1,
2, 1, ... (OEIS A001511). The parity
of this disk-label sequence is given by 1, 0, 1, 1,
1, 0, 1, 0, 1, 0, 1, 1, ... (OEIS A035263)
which, amazingly, also corresponds to the accumulation
point of 
cycles through successive bifurcations.
