zdps VS Fibonacci-Sequence

I think their point is that naive caching attempts still result in duplicate work, and if you start by viewing the problem recursively then DP is a clever insight allowing you to avoid that duplicate work. You're right; it's not _just_ function calling overhead or anything minor; it often represents major Big-O improvements. I think that's in alignment with what they're saying.

I like your insight about Fibonacci needing less state than a naive DP would suggest. Diving into that a bit, DP implementations represent a DAG. More importantly, for all the problems that easily fit into DP that DAG has small numbers of "local" connections -- you can partition it into at least O(some_meaningful_dimension) many topologically sorted levels where each level only depends on a small, known, finite number of previous levels (often 1-2), not deeper DAG nodes. A clever choice of order of operations then looks for a min-cut on that graph, proceeding through those levels. Fib has at most O(n) levels and at least O(n) state for naive DP, so a clever implementation has O(n/n)=O(1) state. Edit distance has at most O(n) levels and at least O(n^2) state, so a clever implementation has O(n^2/n)=O(n) state. I did something similar for some DP probability calculation [0], and it reduces state from quadratic to linear.

Fibonacci is, slightly pedantically, one of the worst algorithms to analyze this way. Only the smallest inputs actually fit in machine integers, and the exponential increase in the size of any bigints involved really puts a damper on a naive Big-O analysis. Fib mod 2^64 (or some biggish prime) fits into the spirit of things better.

[0] https://github.com/hmusgrave/zdps

However, purely recursive solutions can be simpler to understand and implement, especially for problems that are naturally suited to recursion. They can also be more intuitive, as they often closely mirror the problem's definition. For example, consider a recursive function to calculate the Fibonacci sequence. The problem's definition naturally lends itself to a recursive solution: the nth Fibonacci number is the sum of the (n-1)th and (n-2)th Fibonacci numbers.

The Fibonacci sequence is a sequence in which each number is the sum of the two preceding numbers. For example, starting at 0 and 1 the next number in the Fibonacci sequence would be 1. Followed by: 2, 3, 5, 8 and so on. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers. So we're going to have to write a function that detects Fibonacci numbers.

If you have read any articles on the topic of recursion the chances are they used either the Fibonacci Sequence or the calculation of Factorial numbers as the case study. Both of these are rather trivial and somewhat pointless examples outside of their academic purpose. However, to ease in this article we will briefly reprise the examples.

I think you're best off seeking information about it's appearance in history and in Renaissance architecture, in math you need to go in a different direction and look into the people inspired by it, https://en.wikipedia.org/wiki/Fibonacci_sequence is more where Mathematics finds it interesting. In nature it only appears in things like flowers because it's a solution to sphere packing.. That said there's a lot less there than some people would have you believe, should be enough for a paper though!

This article will explain the two variable assignment methods and will use the Fibonacci sequence to illustrate the variation in results caused by each of them.

It’s incredibly useful to grasp concepts you’ve not come into contact with. Let’s take the Fibonacci sequence as an example: https://en.wikipedia.org/wiki/Fibonacci_sequence

Glad you like it! The spiral of the snake follows the Fibonacci sequence.

The golden ratio is a ratio of length and width that is surprisingly pleasant to the human eye, the word "golden" itself doesn't really have a specific meaning. To find the ratio of a width and length closest to the golden ratio (1.618...) you can use two numbers following each other in the Fibonacci Sequence.