This is another Fibonccoid Sequence [a Fibonacci-like Sequence where 2 numbers are added together to give the next
F(n) = F(n-1) + F(n-2)],
meaning that certain numbers are added to one another to generate the next number,
which may be expressed in squares or in this strange case, as triangles.
The Padovan Sequence goes like this:
1 - 1 - 1 - 2 - 2 - 3 - 4 - 5 - 7 - 9 - 12 - 16 - etc.
Take the number 16 above.
How was this number or triangle generated?
In this infinitely additive sequence,
add the 2nd and 3rd number to its left, ignoring the immediate number before the 16:
ie: ignore the 12 and add the sum of 9 + 7 which gives 16.
Thus, the next Padovan Number after 16 would be 12 + 9 = 21.
In Mathematics, we would say that P(n) = P(n-2) + P(n-3)
The number that is skipped or ignored, the number that is immediately before the last Padovan Number must be expressed as P(n-1).
Observe the elegant spiral that is formed when we arc our compass from triangle to triangle,
each triangle getting enlarged, each spiral growing.
This is certainly another unusual nugget in the world of mathematical gems." – Jain 108
F(n) = F(n-1) + F(n-2)],
meaning that certain numbers are added to one another to generate the next number,
which may be expressed in squares or in this strange case, as triangles.
The Padovan Sequence goes like this:
1 - 1 - 1 - 2 - 2 - 3 - 4 - 5 - 7 - 9 - 12 - 16 - etc.
Take the number 16 above.
How was this number or triangle generated?
In this infinitely additive sequence,
add the 2nd and 3rd number to its left, ignoring the immediate number before the 16:
ie: ignore the 12 and add the sum of 9 + 7 which gives 16.
Thus, the next Padovan Number after 16 would be 12 + 9 = 21.
In Mathematics, we would say that P(n) = P(n-2) + P(n-3)
The number that is skipped or ignored, the number that is immediately before the last Padovan Number must be expressed as P(n-1).
Observe the elegant spiral that is formed when we arc our compass from triangle to triangle,
each triangle getting enlarged, each spiral growing.
This is certainly another unusual nugget in the world of mathematical gems." – Jain 108
