Let b(k) be the number of 1s in the binary expression of k, i.e., the binary digit count of 1, giving 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120) for k = 1, 2, .... Then the number of odd binomial coefficients (k j) where 0<=j<=k is 2^(b(k)). This means that the number of odd elements in the first n rows of Pascal's triangle is f_n = sum_(k = 0)^(n - 1) 2^(b(k)), the first few terms of which are 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, ... (OEIS A006046).

batrachion | binary | binomial coefficient | blancmange function | digit count | Hofstadter-Conway $10, 000 sequence | Pascal's triangle | Rudin-Shapiro sequence

Heiko Harborth