Narayana-Zidek-Capell numbers
Ruby, 34 32 bytes

This uses a formula from the OEIS page for the Narayana-Zidek-Cappell numbers.

Edit: Got rid of parentheses using operator precedence with thanks to feersum and Neil.

f=-&gt;x{x&lt;4?1:2*f[x-1]-x%2*f[x/2]}