**Does the Intensity of Binary
Audioactive Decay Converge?**

** **

The *binary look and say sequence* (BLS) is akin to the
well-known look and say sequence. Starting with a 1, the next term is
obtained from the previous term by describing its digits *in binary*. [This
process is called *binary audioactive decay*, after the audioactive
decay studied by J. Conway.]

For example, 1 is described as one 1, so the second term
of the BLS is 11. The second term is described as two 1s, and since 2 = 10_{2},
the third term is 101. Then the third term is described as one 1, one 0,
one 1, so the fourth term is 111011. Next, the fourth term is described as
three 1s, one 0, two 1s, and since 3 = 11_{2} and 2 = 10_{2},
the fifth term is 11110101.

The BLS has first few terms

1, 11, 101, 111011, 11110101, 100110111011, 111001011011110101, .

Now, let o(n) denote the number of ones, and z(n) denote the
number of zeros, in the n-th term BLS(n) of the BLS. For each n > 2, define
the *intensity* i(n) of BLS(n) by the equation

i(n) = o(n)/z(n).

[It is clear that z(n) > 0 for all n > 2.] If the number of 1s are thought of as raising the sound BLS(n), and the number of 0s as decreasing it, then r(n) measures the intensity of the sound BLS(n). The intensities i(n), n = 3, 4, ., 30 are:

2, 5, 3, 2, 2, 2, 25/14, 37/21, 54/31, 79/46, 115/68, 169/100,

247/147, 181/108, 530/317, 259/155, 569/341, 417/250, 1222/733,

3582/2149, 5249/3150, 7693/4617, 11274/6767, 16523/9918, 24215/14536,

35489/21304, 52011/31223, 38113/22880, 37238/22355, 54575/32763,

4799/2881

These intensities are approximately

2, 5, 3, 2, 2, 2, 1.78571, 1.7619, 1.74194, 1.71739, 1.69118, 1.69,

1.68027, 1.67593, 1.67192, 1.67097, 1.66862, 1.668, 1.66712, 1.66682,

1.66635, 1.66623, 1.66603, 1.66596, 1.66586, 1.66584, 1.66579, 1.66578,

1.66576, 1.66575, 1.66574

The behavior of these intensities suggests the following
question:

** **

*Does the intensity i(n)
converge to 5/3 ~ 1.67 or some nearby value?*

* *

Joseph L. Pe

31 January 2003