User blog:Googleaarex/N-Goodstein Sequence

1-Goodstein Sequences - G(n,1)
G(n,1) is same at G(n,3), but uses addition.

G(2,1)
X,2

2,3

1,4

0,5

G(3,1)
X+1,2

X,3

3,4

2,5

1,6

0,7

G(4,1)
X+X,2

X,5

0,11

G(5,1)
X+X+1,2

X+X,3

X,7

0,15

So G(n*2+1,1) = 2^{n+2}-1

G(6,1)
X+X+X,2

X+X,5

X,11

0,23

So G(n*2,1) = 3*2^n-1

2-Goodstein Sequences - G(n,2)
G(n,1) is same at G(n,3), but uses addition and multiplication.

G(5,2)
X*X+1, 2

X*X, 3

X*3+3, 4

X*3, 7

X*2, 15

X, 31

0, 63

G(6,2)
X*(X+1), 2

X*X+2, 3

X*X, 5

X*5+5, 6

X*5, 11

X*4, 23

0, 287

G(7,2)
X*(X+1)+1, 2

X*(X+1), 3

X*X, 7

X*6, 15

0, 1023

G(8,2)
X*(X*X), 2

X*(X*2+2)+2, 3

X*(X*2+2), 5

X*(X*2+1), 11

X*(X*2), 23

X*(X+23)+23, 24

X*(X+23), 47

X*X, 24*2^24-1