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 “1”s, and since 2 = 10₂, 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 “1”s, one “0”, two “1”s, and since 3 = 11₂ and 2 = 10₂, 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 “1”s are thought of as raising the “sound” BLS(n), and the number of “0”s 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