Skew binary number system

Given a skew binary number, its value can be computed by a loop, computing the successive values of ${\displaystyle 2^{n+1}-1}$ and adding it once or twice for each ${\displaystyle n}$ such that the ${\displaystyle n}$th digit is 1 or 2 respectively. A more efficient method is now given, with only bit representation and one subtraction.

The skew binary number of the form ${\displaystyle b_{0}\dots b_{n}}$ without 2 and with ${\displaystyle m}$ 1s is equal to the binary number ${\displaystyle 0b_{0}\dots b_{n}}$ minus ${\displaystyle m}$. Let ${\displaystyle d^{c}}$ represents the digit ${\displaystyle d}$ repeated ${\displaystyle c}$ times. The skew binary number of the form ${\displaystyle 0^{c_{0}}21^{c_{1}}0b_{0}\dots b_{n}}$ with ${\displaystyle m}$ 1s is equal to the binary number ${\displaystyle 0^{c_{0}+c_{1}+2}1b_{0}\dots b_{n}}$ minus ${\displaystyle m}$.

Similarly to the preceding section, the binary number ${\displaystyle b}$ of the form ${\displaystyle b_{0}\dots b_{n}}$ with ${\displaystyle m}$ 1s equals the skew binary number ${\displaystyle b_{1}\dots b_{n}}$ plus ${\displaystyle m}$. Note that since addition is not defined, adding ${\displaystyle m}$ corresponds to incrementing the number ${\displaystyle m}$ times. However, ${\displaystyle m}$ is bounded by the logarithm of ${\displaystyle b}$ and incrementation takes constant time. Hence transforming a binary number into a skew binary number runs in time linear in the length of the number.