# 2^n+1

### 2^n+1

#### Introduction

• We can generate a sequence of integers using the following equation: n 1 2 3 4 5 6 7 8 2n + 1 3 5 9 17 33 65 129 257
• These numbers have an important property when it comes to calculating their midpoints:
• To calculate the midpoint between two numbers, you can use this equation: • For example: the midpoint between 1 and 17 is: • The midpoint between 1 and any number in the sequence 2n + 1 is equal to the previous number in the sequence
• For example:
• The midpoint between 1 and 33 is 17
• The midpoint between 1 and 17 is 9
• The midpoint between 1 and 9 is 5
• Proof:
• the midpoint between 1 and 2n + 1 is: • A number which is in the sequence 2n + 1 can be recursively split into two segments until each segment has a size of 1: #### code (Python)

``````from collections import deque

# The following code creates an array of numbers starting at 0 and smoothly increasing to 257

n = 3

# creates an array of 0's of size 2^n + 1
x = *(2**n + 1)

# set the value for the last element
x[len(x) - 1] = 257

# we create a queue of the line segments we need to subdivide
q = deque()

# and add the first segment to it
q.append((0, len(x) - 1))

# now we go through and subdivide each segment
while len(q) != 0:
left, right = q.popleft()
midpoint = (left + right) // 2

# set the midpoint to the average of the two ends
x[midpoint] = (x[left] + x[right]) // 2

# if the width of the segment is greater than 2 then it can be subdivided
if right - left > 2:
q.append((left, midpoint))
q.append((midpoint, right))

print(x) # prints: [0, 32, 64, 96, 128, 160, 192, 224, 257]``````