Actually, the main reason for this post is my dissatisfaction with their solution and my belief that a more elegant solution can be found..(though most probably i am wrong on this account)

Thanks in advance for such a solution

PONDER THIS

Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each power no more than twice.
For example, f(10)=5 since there are five different ways to express 10: 1+1+8, 1+1+4+4, 1+1+2+2+4, 2+4+4 and 2+8.
Describe, in a single sentence, the multiset {f(n)/f(n-1)} for positive integer n.
Show your proof to this sentence.

Have a Good day
-Akash Kumar

https://www.cut-the-knot.org/blue/Fusc.shtml