The Atone Sequence (Atonen Sequentia)

atonen.md

ATONEN

par monsieur Shimon Degli Ioannis (alias Kolis)

Known as Gaussian binomial coefficient. The Atone Sequence was given this name (unofficially) by Applied Mathematician Victor deMatos. Meaning the sequence of ones. Because when converted to binary the numbers are formed only by 1's:

1, 11, 111, 1111, 11111, 111111 ... ∞ -> {1, 3, 7, 15, 31, 63 ... ad infinitum}

Pascal's Triangle Displaced Degrees

In Pascal's Traite Dv Triangle Arithmetiqve, where he explores the arithmetic of the Triangle. He also calls the horizontal bands -> horizontal rows, and the vertical ones, perpendicular rows. Alongside with the triangle, Victor deMatos believes there are two other triangles on either side. Representing the exponential expression, minus one.

1 - Pascal's Triangle

     1
    1 1
   1 2 1
  1 3 3 1
 1 4 6 4 1
ad infinitum

If you were to add up the rows, you get 2ⁿ.

1 -> (no addition, meaning 0, zero) = 2⁰
1 + 1 -> 2 = 2¹
1 + 2 + 1 -> 4 = 2²
1 + 3 + 3 + 1 -> 8 = 2³
1 + 4 + 6 + 4 + 1 -> 16= 2⁴

Also (unofficially) called 2-11 relationship.

2 - Shim'n Degli Ioannis Pascal's Displaced Degrees Triangle (alias Kolisian Triangles)

     1
    1 1
   1 1 1
  1 1 1 1
 1 1 1 1 1
ad infinitum

If you were to add up the rows and subtract one, you get 2ⁿ - 1.

1 -> 1 = 2¹ - 1
1 + 1 -> 2 = 2² - 1
1 + 2 + 1 -> 4 = 2³ - 1
1 + 3 + 3 + 1 -> 8 = 2⁴ - 1
1 + 4 + 6 + 4 + 1 -> 16= 2⁵ - 1

Juxtaposition

 1 1 1 1 1 1 1 1 1
  1 1 1 1 1 1 1 1
   1 1 1 2 1 1 1
    1 1 3 3 1 1
     1 4 6 4 1

Finiting the juxtaposition (ad finitum).

 1 1 1 1 1 1 1 1 1
  1 1 1 1 1 1 1 1
   1 1 1 2 1 1 1
    1 1 3 3 1 1
     1 4 6 4 1
      1 3 3 1 
       1 2 1
        1 1
         1

For simplicity's sake, the triangle can be called, Pascal's Displaced Triangle or The Atone(n) Triangle.

(contribute, https://github.com/shimon-d/functvs)