# 3 More Amazing Math Sequences Beyond Fibonacci

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The Fibonacci sequence is one of the most interesting and perhaps mysterious topics in Math. Just how many sequences do you have which is related to rabbit reproduction? Yet, there are still more amazing math sequences you might not know and here's three of them.

The Lucas Numbers

Mathematicians refer to the Lucas numbers as the Fibonacci's sibling. Extensively studied by François Édouard Anatole Lucas, the Lucas number sequence goes like this: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207 and it goes on and on.

Some of the reasons why the Lucas numbers are amazing is that first, it cannot be divided by 5 or 13; second, it cannot be divided by a Fibonacci number except by 1,2, and 3; third, it follows a 12-cycle length pattern. If you look at the given sequence above, it repeats the sequence after the 12th number in the sequence.

Yellowstone Permutation Integer Sequence

According to Owlcation, this sequence got its name because its permutation looks like the geysers found at the Yellowstone National Park when you look at it on a graph. Moreover, the entire sequence is a permutation of all the positive integers in the sequence. And this is what makes it one of the amazing math sequences.

The Yellowstone sequence follows the rule where the fourth number in the sequence should have a common factor with 2 but not with 3; the fifth number should have a common factor with 3 but not with 4; and the sixth number should have a common factor with 4 but not with 9. Thus, the sequence will look like this: 1, 2, 3, 4, 9, 8, 15, 14, and so on.

Both these sequences are nicknamed "skiponacci" because the sum of the next number in the sequence is the sum of the two previous numbers within the sequence. The only difference is that the Padovan sequence starts with 1, 1, 1 while the Perrin sequence starts with 3, 0, 2.

A Padovan sequence will look this: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37 and so on.

The Perrin sequence, on the other hand, will look like this: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158 and so on.

What makes them amazing math sequences is that both of their cycle repeats after the 168th number in the sequence, and the ratio between the consecutive terms approach a limiting value called a plastic constant. In mathematics, a plastic constant is an irrational number and the real-valued root of the cubic equation x^3 - x - 1 = 0.