Explore a Klarner-Rado sequence

05AB1E, 15 14 bytes

$FDx>DŠ+)˜ê}sè

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JavaScript (ES6),  53  50 bytes

1-indexed.

n=>(g=k=>n?g(g[k]|!k++?g[n--,k*2]=g[k*3]=k:k):k)``

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Commented

Note: On the first iteration, we have k = [''], which is zero-ish but truthy. By doing !k++, we force k to be coerced to \$0\$ right away (and not just before it's incremented), which makes the test work as expected.

n => (                    // n = requested index
  g = k =>                // h is a recursive function taking k, starting at 0
    n ?                   // if n is not equal to 0:
      g(                  //   do a recursive call:
        g[k] |            //     if g[k] is defined
        !k++ ?            //     or k = 0 (increment k after the test):
          g[n--, k * 2] = //       decrement n; set g[k * 2]
          g[k * 3] = k    //       and g[k * 3] and pass k
        :                 //     else:
          k               //       just pass k
      )                   //   end of recursive call
    :                     // else:
      k                   //   stop recursion and return k
)``                       // initial call to g with k = [''] (zero-ish)

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JavaScript (ES6), 65 bytes

I thought the '11'[k/x-2] trick was neat, but overall this initial approach is far too long.

0-indexed.

n=>(g=k=>a[n]||g(-~k,a.some(x=>'11'[k/x-2])&&a.push(k+1)))(a=[1])

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Commented

n => (                  // n = requested index
  g = k =>              // g is a recursive function taking k (starting at 1)
    a[n] ||             // if a[n] is defined, return it and stop
    g(                  // otherwise, do a recursive call:
      -~k,              //   with k + 1
      a.some(x =>       //   if there exists some x in a[]
        '11'[k / x - 2] //   such that k / x is either 2 or 3 ...
      )                 //
      && a.push(k + 1)  //   ... then push k + 1 in a[]
    )                   // end of recursive call
)(a = [1])              // initial call to g with k = a = [1]