Gould's sequence

Gould's sequence is an integer sequence named after Henry W. Gould that counts the odd numbers in each row of Pascal's triangle. It consists only of powers of two, and begins:

For instance, the sixth number in the sequence is 4, because there are four odd numbers in the sixth row of Pascal's triangle (the four bold numbers in the sequence 1, 5, 10, 10, 5, 1).

The nth value in the sequence (starting from n = 0) gives the highest power of two that divides the central binomial coefficient ${\displaystyle {\tbinom {2n}{n}}}$, and the numerator of ${\displaystyle 2^{n}/n!}$ (expressed as a fraction in lowest terms).

Gould's sequence also gives the number of live cells in the nth generation of the Rule 90 cellular automaton starting from a single live cell. It has a characteristic growing sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90.

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