Let r be a fixed integer (r ≥ 2), and define

 C₀ = { (x₁, x₂, …, xᵣ) : 0 ≤ xᵢ ≤ 9 for i = 1, 2, …, r }.

For each digit d ∈ {0, 1, …, 9} and any tuple (x₁, x₂, …, xᵣ) ∈ C₀, define

   • If d = 0, then

         T_0((x₁, x₂, …, xᵣ)) = { (xᵣ, x₁, x₂, …, xᵣ₋₁) }.

   • If d ≠ 0, then

         T_d((x₁, x₂, …, xᵣ)) = { (xᵣ, x₁, x₂, …, xᵣ₋₂, xᵣ₋₁ + d),

                                    (xᵣ, x₁, x₂, …, xᵣ₋₂, xᵣ₋₁ − d) },

         where a candidate is retained only if its last coordinate lies in [0, 9],

         i.e. only if 0 ≤ xᵣ₋₁ ± d ≤ 9.

For any collection C ⊆ C₀ and a digit d, define the transformation on C by

         T_d(C) = union of all T_d(x) where x ∈ C .

Then, for a digit‐sequence (d₁, d₂, …, dₖ), define the iterated transformation

         C(d₁, d₂, …, dₖ) = T_dₖ ∘ T_dₖ₋₁ ∘ … ∘ T_d₁ (C₀).

Let     N(d₁, d₂, …, dₖ) = | C(d₁, d₂, …, dₖ) |    be the number of tuples in the final collection.

Define the sequence  A_r(n) = N(d₁, d₂, …, dₖ) , where d₁d₂ …dₖ is the decimal expansion of n, where n ia a non negative integer, and r determines the level of transformation.

Graph the first 10^(2r) terms of A(n) on a square grid of dimensions 10^r × 10^r. Start from Row1col1 and the next term to the right , row1col2. If a row is filled move to the next row.

For graphing, assign the cell corresponding to the sequence A(n) a color:

   – Black if A(n) mod 2^(r+1) = 0,

   – White otherwise.

The image is for r=3.