## Two Parallel Recurrences

Two sequences \(a(0), a(1), a(2), \ldots \) and \(b(0), b(1), b(2), \ldots \) are defined by:

- \(a(0) = b(0) = 0\),
- \(a(n+1) = a(n) + b(n) + 1\), and
- \(b(n+1) = a(n) + 2b(n) + 3\).

Part A | What is \(a(10)\)? | |
---|---|---|

Part B | What is the remainder when \(a(2021)\) is divided by \(10^9+7\)? | |

Part C | What is the remainder when \(a(1234567890)\) is divided by \(10^9+7\)? |

## Inverse Harmonic Function

The *Harmonic Series* \(H(n)\) is defined to be the sum of the reciprocals of
the first \(n\) positive integers.
\( H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} \).
The *Inverse Harmonic Function* \(I(x)\) is defined to be
the smallest positive integer \(n\) such that \(H(n) \geq x\).
So for example \(I(2.0) = 4\) because \(H(4) > 2.0\) but \(H(3) < 2.0\).

Part A | What is \(I(4)\)? | |
---|---|---|

Part B | What is \(I(18)\)? |