A powerfully, powerfully, powerfully written number sequence

Find the missing terms a17 , a19 and a22 of the sequence,

15 , 20 , 13 , 2 , 20 , 17 , 17 , 19 , 19 , 15 , 14 , 4 , 21 , 4 , 20 , 17 , a17 , 19 , a19 , 15 , 14 , a22

The sequence uses only basic Mathematics and less common English.

• curious why you removed the last term? – tilper Aug 29 '17 at 15:31
• Good spot - it was incorrect - you may see why from a certain table . The sequence can be extended indefinitely (and I believe uniquely using a convention given by two English mathematicians). The earlier terms in the sequence should, I hope, be enough to identify it, after using the applied tags. – Tom Aug 29 '17 at 15:39

The values are

a17 = 17, a19 = 19, a22 = 22

The sequence might be easier to see if we first

Use A1Z26 to convert all elements to letters: o , t , m , b , t , q , q , s , s , o , n , d , u , d , t , q , ? , s , ? , o , n , ?

Which can then be seen to mean:

The powers of one thousand: one, thousand, million, billion, trillion, quadrillion, quintillion, ...

The missing elements are Quinquadecillion, Septendecillion and Vigintillion, which converted back to numbers are 17, 19 and 22 respectively.

The values are:

a17 = 17, a19 = 19, a22 = 4

This is because...

there is a loop in the graph. Notice how points 5 to 12 are a complete version of points 15 to 22.

• i don't think that's going to be the answer – JMP Aug 31 '17 at 4:16
• Could you give some follow up to your statement? I'm not disagreeing with you, but would like your reasoning. – wyskoj Aug 31 '17 at 4:18
• i was working on an interpolated polynomial – JMP Aug 31 '17 at 4:20
• Ah, that's a good idea. – wyskoj Aug 31 '17 at 4:22
• ... or a Fourier series? – Tom Aug 31 '17 at 8:31