## Wednesday, December 9, 2015

### Balance scale series

Trying to get the least amount of elements in a series of natural numbers that can be combined by adding or subtracting unique elements of the series to get as many consecutive natural numbers as possible starting with 1. I came up with:

1,2,7, 21, 52
For example
13=21-7-1,
14=21-7,
15=21-7+1, etc... Every natural number less than 84 can be reached by adding or subtracting these first 5 numbers...

The pattern I found is that x-sum(previous) = sum(previous)+1, solve for x, because the previous were able to reach every number up to sum(previous), and you need an x so that subtracting all of those numbers yields every number up to x...

This isn't in the Encyclopedia of Integer sequences yet, but I don't know if it's the best solution...
The name comes from the fact that on a balancing scale, you could weigh any natural number weight by putting the additions on one side of the balance and the subtractions and the object being weighed on the other...