= Pascals Triangle =


== Construction ==

Pascals triangle is constructed via staggering numbers in rows, and making each
entry equal to the sum of the two entries above it.

For some entry `(i,j)` where `i` is the row and `j` is the column, the value
can be calculated via

`f(i,j) = f(i-1, j-1) + f(i-1,j)`

== Patterns ==

* The sum of elements in a row is twice the sum of the row proceeding it. Or in
  other words, the sum of the elements in row `n` is `2^n`.
* The sum of the square of the elements of row `n` is equal to the middle
  element of row `2n`
* For some row `n` where `n` is prime, all terms in that row except for the 1s
  are multiples of `n`
* To count odd terms in a row `n`, convert `n` to binary. Let `x` be the number
  of 1s in the binary value (popcnt) Then the number of odd terms will be `2^x`
  These values are known as [[goulds_sequence]]
* Every entry in row `2^(n-1) for n>=0` is odd.
* Diagonals along left and right edges contain only 1s
* Diagonals next to the edge of the diagonals count upward
* Next diagonal inward contains the [[triagnular_numbers]]