TAOCP 3.2.2 Exercise 20
A suitable choice for CONTENTS$(A)$ in method (10) is any $k$-bit binary number in which roughly half of the bits are $0$ and half are $1$.
Exercise 20. [**] $[M40]$ Calculate constants suitable for use as CONTENTS$(A)$ in method (10), having approximately the same number of zeros as ones, for $2 \le k \le 64$.
Verified: no
Solve time: 2m01s
A suitable choice for CONTENTS$(A)$ in method (10) is any $k$-bit binary number in which roughly half of the bits are $0$ and half are $1$. For example, one can alternate $0$ and $1$ in the $k$-bit word, or select a pattern derived from a small segment of a maximum-length additive sequence modulo 2, such as the least significant bits of a lagged Fibonacci sequence $(X_n = X_{n-24} + X_{n-55} \bmod 2)$, truncated to $k$ bits. These choices ensure that approximately the same number of zeros and ones appear, which is desirable to prevent initial bias in the generated sequence.
Explicitly, for each $k$ from $2$ to $64$, one may define CONTENTS$(A)$ as the integer whose binary expansion is
$(1010\ldots)_2 \text{ if } k \text{ is even}, \qquad (1010\ldots 1)_2 \text{ if } k \text{ is odd},$
or equivalently, as the integer $\sum_{i=0}^{k-1} 2^i , b_i$, where $b_i = 0$ if $i$ is even and $b_i = 1$ if $i$ is odd. This construction gives exactly $\lfloor k/2 \rfloor$ ones and $\lceil k/2 \rceil$ zeros when $k$ is even, and vice versa when $k$ is odd, satisfying the requirement for approximately equal numbers of zeros and ones. Any other pattern achieving the same approximate balance is also valid.
$$ \boxed{\text{CONTENTS}(A) = (1010\ldots)_2 \text{ for each } k, \text{ with roughly equal zeros and ones.}} $$