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=->x{x<4?1:2*f[x-1]-x%2*f[x/2]}

Jelly, 11 10 bytes

HĊrµṖ߀Sȯ1

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Takes n as argument and prints the result.

Explanation

H              divide input by 2
 Ċ             round up to get first n to recurse
  r            inclusive range from that to n
   µ           (chain separator)
    Ṗ          remove n itself from the range
     ߀        call self recursively on each value in the range
       S       sum results
        ȯ1     if sum was zero, return one

Python 2, 48 42 38 36 bytes

Algorithm taken from the OEIS page. n<3 may be changed to n<4 with no effect. Returns the nth number, where n is a positive integer.

a=lambda n:n<3or 2*a(n-1)-n%2*a(n/2)

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