CPhill: If I haven't forgotten it!!. You start with the last number 6. You take the reciprocal of 6, which is:0.16666666, then add to it the next to last number, which is 1 and you get 1.1666666. Then multiply this by the last number or 6, which will give you 7. Then take 1.1666666... and take its reciprocal which will give you 0.857142........, then add the 3rd from the last which is a 2 and you get
2.857142......... and then multiply this by the last whole number we got, which was 7 and you should get 20. Then you would take the reciprocal of this last 2.857142 which will give you 0.35, then add the next number from the end which is a 1 and you get 1.35. Then multiply this by the last whole integer we got, which was 20 and you will get 27. and so on until you get 229 as numerator and the last step should give you 330 which is the denominator.
Of course, the opposite of this is to obtain the integers of the continued fraction such as we had here, i.e. 0.69393939393. You would take the reciprocal of this and you would get 1.4410480349... and you subtract the integer part, which is a 1. Then you will take the reciprocal of the remaining fraction or, 0.4410480........and subtract the integer part, which a 2........and so on. That is how W/A gets those numbers, i.e. [0; 1, 2, 3, 1, 2, 1, 6].
P.S. I have programmed this into my computer which does it almost intantaneously. I don't know if you know how to program, which you could do easily. Even if you are familiar with spreadsheets like Excel, you could program it into it and it will automate this whole process very rapidly.
