how to find number of presentation of 1 as a sum of $2^{-i}$?

Asked Mar 26 '23 at 14:27

Active Mar 26 '23 at 19:46

Viewed 104 times

How to find number of presentations of 1 as a sum of exactly $k$ numbers of the form $2^{-i}$ ? As an example for $k=2$ we have only one presentation: $$1 = \frac{1}{2} + \frac{1}{2},$$ so answer for $k = 2$ is 1. For $k=4$, $$1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8}$$ and $$1 = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4},$$ so the answer is 2.

How to find the formula for all $k$?

edited Mar 26 '23 at 19:46

JBL

asked Mar 26 '23 at 14:27

rrryok

1

So you me sum of powers of $2^{-1}?$ – Thomas Andrews Mar 26 '23 at 14:29
sorry, yes i mean sum – rrryok Mar 26 '23 at 14:31
Hint I haven't tried. There's a recursion lurking here since with $k$ terms you can expand one of them to find an expression with $k+1$ terms. Sometimes the expansions will match, which you have to account for, – Ethan Bolker Mar 26 '23 at 14:35
1

Generating function, but it doesn't seem to get us anywhere. Consider:
$$f_m(w,z)=\prod_{j=0}^{m}\frac1{1-wz^{2^{j}}}$$

Then you want to find the coefficient of $w^kz^{2^m}$ for $m=\lfloor \log_2 k\rfloor.$ But I don't immediately see how to do that.
– Thomas Andrews Mar 26 '23 at 14:39
That wasn't my question. Reread my first comment. $1=\frac12+\frac12$ is the only way to write $1$ as the sum of $2^{-1}.$ $2^{-2}$ is not $2^{-1}.$ @rrryok – Thomas Andrews Mar 26 '23 at 14:41
2

But ... @rrryok ... How is $1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}$ a sum of $2^{-1}$'s? – Mar 26 '23 at 14:48
You actually can use the infinite product of my formula for the general formula. Interesting, in the infinite product, the sequence of coefficients of $w^kz^{2^m}$ becomes constant for $m$ large, and that constant is the value you want. In particular, we can always choose $m=k.$ – Thomas Andrews Mar 26 '23 at 14:50
3

This is https://oeis.org/A002572 – JBL Mar 26 '23 at 15:12
@Thomas Andrews, @ Stinking Bishop, sorry, my bad, there is $2^{-i}$ – rrryok Mar 26 '23 at 16:19
@Thomas Andrews sorry, i don't get what it's w and z in your formula, can you give an example in usage – rrryok Mar 26 '23 at 17:41
1

This came up again today: Find a formula for this sequence: 1,1,1,2,3,5,9,16,28,50,89... – MJD Mar 26 '23 at 20:43
1

@mick No, $m\neq w.$ I was giving a generating function, which, if you don't know generating functions, probably isn't going to be of much use to you. – Thomas Andrews Mar 26 '23 at 22:47
@ThomasAndrews oh sorry i thought the product went to infinity and was not seeing the m. apologies – mick Mar 26 '23 at 23:02
Think binary is the easiest. – mick Oct 23 '23 at 11:12

how to find number of presentation of 1 as a sum of $2^{-i}$?

0 Answers0

Linked