Calculate the nth term of Golomb's self-describing sequence

Jelly, non-competing

10 bytes This answer is non-competing, since the challenge predates the creation of Jelly.

’ßßạ¹ß‘µṖ¡

This uses the recursive formula a(1) = 1, a(n + 1) = 1 + a(n + 1 - a(a(n))) from the OEIS page.

Try it online!

How it works

’ßßạ¹ß‘µṖ¡ Main link. Input: n

’          Decrement; yield n - 1.
 ßß        Recursively call the main link twice, with argument n - 1.
   ạ¹      Take the absolute difference of a(a(n - 1)) and n.
     ß     Recursively call the main link, with argument n - a(a(n - 1)).
      ‘    Increment the result, yielding 1 + a(n - a(a(n - 1))).
       µ   Combine the chain to the left into a single link.
        Ṗ  Pop [1, ..., n]. This yields [] iff n == 1.
         ¡ Execute the chain iff Ṗ returned a non-empty array.

GolfScript (31 chars)

~([1 2.]2{.2$=[1$)]*@\+\)}3$*;=

Demo

PHP - 63 Chars

function g($n){for(;++$i<=$n;){for(;++$j<=$i;){echo $i;}$j=0;}}

Fast AND short.

I appear to have had the wrong sequence in mind. Derp.

This is CORRECT, fast, and short.

function g($n){for(;++$i<$n;){echo round(1.201*pow($i,.618));}}

Accuracy may suffer past the required 100,000 mark, but I did in fact meet the mark.