Is there an equation for this set of numbers? How did you come up with it?

Question

x y
-------
0 0
1 1
2 3
3 7
4 15
5 31
6 63

The Domain and Range appear to be all positives.

As a bonus, can anyone tell me where I got this sequence??

Ok, I'm slightly embarrased that I didn't see the pattern. I spent some time trying to come up with an equation by calculating a couple of points. Anyone know of a way to do that?

btw, the sequence comes from the Tower of Hanoi puzzle: x is the number of discs, and y is the fewest number of moves needed to complete the puzzle.

10 points still available if anyone can derive an equation from the numbered pairs only, like I was trying to do originally.

Answer 1

This is the Gaussian binomial coefficient [n,1] for q=2.

Greetings from The On-Line Encyclopedia of Integer Sequences!

Answer 2

3

Answer 3

You are always adding the next power of 2 to the previous number.

Power of 2: 1, 2, 4, 8, 16, 32...

0 + 1 = 1
1 + 2 = 3
3 + 4 = 7
7 + 8 = 15
15 + 16 = 31
etc

Answer 4

y = (2^x) - 1

2^0 - 1 = 1 - 1 = 0
2^1 - 1 = 2 - 1 = 1
2^2 - 1 = 4 - 1 = 3
2^3 - 1 = 8 - 1 = 7
2^4 - 1 = 16 - 1 = 15
2^5 - 1 = 32 - 1 = 31
2^6 - 1 = 64 - 1 = 63

Answer 5

0 0
1 1
2 3--->1x2+1=3
3 7 --->3x2+1=7
4 15--->7x2+1=15
5 31--->15x2+1=31
6 63---->31x2+1=63

Answer 6

surely it's just a simple progression where the left column increases by 1 and the right = the left added to the sucessive right column numbers so in the next lines
7 117
8 245

Answer 7

For a given x, y describes the largest number expressible in binary using x digits.

Answer 8

y = (2^x) - 1

I came up with it by noticing the pattern: the powers of two (for each exponent 'x') are all 1 greater than your given corresponding 'y' values.

Answer 9

x = xsub(x-1) + 1
y = sum(ysub1..ysubx) + x

Came up with it by observing pattern in numbers.

No clue on bonus; assuming you made it up?

Answer 10

xyy(n)-y(n+1)2^(x-1)
00
1111
2322
3744
41588
5311616
6633232

y(n+1)=y(n)+2^(n-1)