Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are 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). Gould's sequence is also a fractal sequence.

The nth value in the sequence (starting from n = 0) gives the highest power of 2 that divides the central binomial coefficient ${\displaystyle {\tbinom {2n}{n}}}$, and it gives 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.

The binary logarithms (exponents in the powers of two) of Gould's sequence themselves form an integer sequence,

in which the nth value gives the number of nonzero bits in the binary representation of the number n, sometimes written in mathematical notation as ${\displaystyle \#_{1}(n)}$. Equivalently, the nth value in Gould's sequence is

Taking the sequence of exponents modulo two gives the Thue–Morse sequence.

The partial sums of Gould's sequence,