Yes, the most common one is the sunflower.
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Leonardo Fibonacci, or Leonardo of Pisa.
Fibonacci found a way to present mathematical numbers so that each number in the sequence is the sum of the two previous numbers. For example, if the sequence starts at 0 and 1, then next number in the sequence is 1, the next number would be 2, and then the next number would be 3, and then 5.
The Fibonacci sequence is actually one of the most used mathematical ideas in the world! This is because the differences between the terms of the sequence give birth to the Golden Ratio (which is about 1:1.62). This 'magic' ratio is often said to hold the key to beauty, and is found throughout the natural world - including in the ratio of sizes in a human face. Today it is most commonly used by architects, artists, fashion designers and the like - for anyone that looks to create beauty the Fibonacci Sequence and the Golden Ratio is vital because it tells you how the size you need to make things to make them visually appealing.
add each number to the number before it to get the next number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on The sequence is used to express that which is pleasing to the eye. when displayed in squares it creates a spiral form. The ratios of any subsequent numbers divided by each other will get you 2.666 repeating or .66666 repeating. 2/3 basically. That ratio is found in nature, attractive peoples faces, etc. Basically anything pleasing to human senses is found to have this ratio. Tool uses it in a lot of their music, and a video on youtube can be found about it. So it really is everywhere. As far as the mathematical proofs and equations, I do not know them.
If I understand your question, you are asking what kind of sequence is one where each term is the previous term times a constant. The answer is, a geometric sequence.