Understanding the All 1's Sequence
The All 1's sequence is a simple sequence where every term is equal to 1. When discussing the idea of adding this sequence "up" and "down," we can interpret these terms mathematically in the context of sequences.

Adding the Sequence Up
When we add the All 1's sequence up, we are essentially creating a cumulative sum:
- **Cumulative Sum**: The nth term of the sequence will be the sum of the first n terms.
- **Formula**: For the All 1's sequence, the nth term is given by:
- T(n) = 1 + 1 + ... + 1 (n times) = n
- **Result**: The resulting sequence is the natural numbers:
- 1, 2, 3, 4, ...

Adding the Sequence Down
When we add the All 1's sequence down, we are subtracting from the cumulative sum. This can be viewed as starting from a certain number and decrementing:
- **Cumulative Subtraction**: If we start from n and subtract 1 for each term, the nth term will be:
- T(n) = n - 1 - 1 - ... - 1 (n times) = 0 (for n=1) or negative integers.
- **Result**: The resulting sequence from this approach is:
- 0, -1, -2, -3, ...

Conclusion
By analyzing the All 1's sequence through these two operations, we derive two distinct sequences: the natural numbers when adding up and the non-positive integers when adding down. This demonstrates the flexibility of sequences in mathematical interpretation.