Fibonacci Variants

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

[worldnotworld]

Thank you. It's fascinating that variations of the Fibonacci series always seem have the F-series "within" them; it has indeed an elemental quality. The first variant, the product variant, looks profoundly different, though, in terms of behavior, with a limit of zero for a, b<1, a fixed point 1 for a=b=1, and infinity for a, b>1... but what if one is >1 and that other <1? or (gasp) negative? I'm going to have to explore what fixed points and limits there are for various a, b. Or perhaps Dr. Barker can to show further properties of these series!

[pmmeurcatpics]

Nice video! It was very satisfying to see the proof to come together at the end

[f-th]

7:38 the sum at the end looks like a convolution, so maybe it would be useful to use generating functions.
It looks like the product of 1/(1-x) with the generating function for the Fibonacci numbers (shifted by x or 1/x)

[sabotagedgamerz]

I’ve studied other Fibonacci style sequences in the form U_n+2 = a*U_n+1 + b*U_n. Where a and b are constants. I found their respective “golden ratios” and their nth term formulas. It was really fun.

[danielmilyutin]

I usually solve unhomogenous equatioon as it is done with ODE. I believe it is called variation of constant method.
We search unhomohenous soution in form:
u_n= A_n*F_n+B_n*F_{n-1}
Putting all that stuff into equation will lead to lots of cancellations.
And equations for A_n, B_n will look like simple integration equations (have to):
A_{n} = A_{n-1} + someG_{n}
B_{n} = B_{n-1} + someH_{n}

[holyshit]

I prefer generating functions (ordinary or exponential) to solve this equation
Z transform also can be useful
Lets try another sequence
u_{0}=1
u_{n+1}=sum(u_{k}F(n-k), k=0..n)
where F(n) is shifted Fibonacci sequence
with F(0)=1, F(1)=1, F(n+2)=F(n+1)+F(n)
Generating function for sum of Fibonacci sequence is
S(x)=x/((1-x)(1-x-x^2))
The first sequence may be problematic for generating functions

[yuyu-mmpk]

Im Japanese High school student but your video is amazing!
Subscrived.

[MrRyanroberson]

here's an idea: are there any simple sequences where the closed formula is something like... the product of fibonacci numbers? or is such a product always reducible to some other fibonacci number?

[hypebeastuchiha]

I like your channel.
Any plans for more videos on statistics?