## Bumerang

### Info

Denote the elements of the field `$F_{4}$`

by `${0, 1, w, w + 1}$`

, where w satisfies the following equation with coefficients in `$F2: w^{2} + w + 1 = 0$`

.
Bumerang is a recurrent double sequence defined by `$a(i, 0) = a(0, j) = 1$`

and `$a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j) )$`

,
where `$f(x, y, z ) = w x^{2} + x + w y^{2} + w z^{2} + (w + 1) z$`

.
This recurrent double sequence can be also obtained using a system of substitutions of type 4 -> 8 with 16 rules, as it follows.

### Substitution Rule

### Patch