The Baum-Sweet Sequence
JavaScript (ES6), 70 68 63 bytes



g=n=&gt;n?g(n-1).concat(/0/.test(n.toString(2).split`00`)?[]:n):[]

console.log(g(1000).join(", "))




Slightly more interesting recursive solution:

n=&gt;[...Array(n+1).keys()].filter(f=n=&gt;n&lt;2?n:n%4?n&amp;f(n&gt;&gt;1):f(‌​n/4))


67 bytes thanks to @Neil.

g is the function to call.