I am trying to understand the following integer sequence because it is related to a project of mine: https://oeis.org/A007178
The sequence is defined as "[The] number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats." My issue with this definition is, if I'm not misunderstanding it, it is too broad. I can interpret it two ways: either the number of ways to write 1 using sums of (1/2)^n or the number of ways to write 1 using n different powers of (1/2). Neither of these are correct as the example for n = 3 is (1/2 + 1/4 + 1/4), (1/4 + 1/2 + 1/4), and (1/4, 1/4, 1/2).
I cannot extrapolate what the permutations for a higher n would like just from this information. Should the sums contain every power of 1/2 up to n? Consider n = 4, the sum
(1/2) + (1/4) + (1/8) + (1/8) can be permuted 4!/2!, or 12 ways.
The sum (1/2) + (1/8) + (1/8) + (1/8) + (1/8) can be permuted 5!/4!, or 5 ways.
I have just listed 17 ordered sums of n powers that equal 1, but the sequence says it should be 13.
Can someone write out what n = 4 would look like?
The sequence counts the number of ways to count sums of n powers of 1/2, as in there are n terms in the sum. Thus, a4 would have all permutations of {1/2, 1/4, 1/8, 1/8} as well as {1/4, 1/4, 1/4, 1/4}.
Thanks to lulu and stuart stevenson for clarification.