Gould's sequence

The nth value in the sequence (starting from n = 0) gives the highest power of 2 that divides the central binomial coefficient

[1][3] It has a characteristic growing sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90.

[4] 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

[5] The partial sums of Gould's sequence, count all odd numbers in the first n rows of Pascal's triangle.

Because of this doubling construction, the first occurrence of each power of two 2i in this sequence is at position 2i − 1.

[3][8][9] In Gould's sequence, the values at odd positions are double their predecessors, while in the sequence of exponents, the values at odd positions are one plus their predecessors.

The sequence is named after Henry W. Gould, who studied it in the early 1960s.

[10][11] Proving that the numbers in Gould's sequence are powers of two was given as a problem in the 1956 William Lowell Putnam Mathematical Competition.

Pascal's triangle, rows 0 through 7. The number of odd integers in row i is the i -th number in Gould's sequence.
The self-similar sawtooth shape of Gould's sequence
Sierpinski triangle generated by Rule 90 , or by marking the positions of odd numbers in Pascal's triangle . Gould's sequence counts the number of live cells in each row of this pattern.