We explore 3 variants on the Fibonacci numbers, focussing on how each sequence is related to the Fibonacci numbers. The main problem is expressing, in terms of the Fibonacci numbers, the nth term of a sequence with recurrence relation u_{n+2} = u_{n+1} + u_{n} + n, and u_{1} = a, u_{2} = b. We also do the same for the recurrence relations u_{n+2} = u_{n+1} u_{n}, and u_{n+2} = u_{n+1} + u_{n} + c.
00:00 Intro
00:11 Example 1
02:12 Example 2
04:24 Example 3
07:45 Simplifying the sum
09:50 Partial sums of Fibonacci numbers
13:02 Proof by induction
