FANDOM


This is FGH version of Dollar Function.

Bracket Notation

1 has growth rate 0 (equal!)

[0] has growth rate 1 (equal!), because n$[0]# = n$n#

[1] has growth rate 2 (equal!), because [@+1] = [@]...[@].

[n] has growth rate n+1 (equal!)

[[0]] has growth rate w, because it nested.

[1[0]] has growth rate w+1, because [@+1 #] = [@ #]...[@ #].

[n[0]] has growth rate w+n

[[0][0]] has growth rate w*2, because [[0] #] = [w #]

[[0]...[0]] (n [0]'s) has growth rate w*n

[[1]] has growth rate w^2

[[1]...[1]] (n [1]'s) has growth rate w^2*n

[[1]] has growth rate w^3

[[n]] has growth rate w^{n+1}

[[[0]]] has growth rate w^w

[[[1]]] has growth rate w^w^2

[[[n]]] has growth rate w^w^{n+1}

[[[[0]]]] has growth rate w^w^w

[...[0]...] (n nested) has growth rate w^^{n-1}

Extended Bracket Notation

[[0]2] has growth rate e(0), because f([0]2) = f([...f([0])...]).

[n[0]2] has growth rate e(0)+n

[[0][0]2] has growth rate e(0)+w

[[[0]][0]2] has growth rate e(0)+w^w

[[...[0]...][0]2] (n nested) has growth rate e(0)+w^^n

[[[0]2][0]2] has growth rate e(0)*2

[[[0][0]2][0]2] has growth rate e(0)*w

[[[[0]2][0]2][0]2] has growth rate e(0)^2

[[[[[0]2][0]2][0]2][0]2] has growth rate e(0)^e(0)

[[...[[0]2]...][0]2] (n*2 nested) has growth rate e(0)^^n

[[...[[0]2]...][0]2] (n*2+1 nested) has growth rate e(0)^...^e(0)^2 (n e(0)'s)

[[0]2[0]2] has growth rate e(1)

[[0]2...[0]2] (n [0]2's) has growth rate e(n-1)

[[1]2] has growth rate e(w), because [@+1 #]@ = [@ #]@...[@ #]@.

[[0]2...[0]2[1]2] (n [0]2's) has growth rate e(w+n)

[[1]2[1]2] has growth rate e(w*2)

[[1]2...[1]2] (n [1]2's) has growth rate e(w*n)

[[2]2] has growth rate e(w^2)

[[n]2] has growth rate e(w^n)

[[[0]]2] has growth rate e(w^w)

[[...[0]...]2] (n nested) has growth rate e(w^^n)

[[[[0]2]]2] has growth rate e(e(0))

[[[[[[0]2]]2]2] has growth rate e(e(e(0)))

[[[...[0]2]...]2] (n 2's) has growth rate e(...(e(0)...) (n nested)

[[[0]2]2] has growth rate z(0)

[[0]2[[0]2]2] has growth rate e(z(0)+1)

[[1]2[[0]2]2] has growth rate e(z(0)+w)

[[[[0]2]]2[[0]2]2] has growth rate e(z(0)+e(0))

[[[[[0]2]2]]2[[0]2]2] has growth rate e(z(0)*2)

[[[0]2]2[[0]2]2] has growth rate z(1)

[[1[0]2]2] has growth rate z(w)

[[[[0]2][0]2]2] has growth rate z(e(0))

[[[[[0]2]2][0]2]2] has growth rate z(z(0))

[[[0]2[0]2]2] has growth rate eta(0)

[[[1]2]2] has growth rate phi(w,0)

[[[[[0]2]]2]2] has growth rate phi(e(0),0)

[[[[0]2]2]2] has growth rate gamma(0)

[[[[0]2[0]2]2]2] has growth rate phi(1,0,0,0)

[[[[0]2...[0]2]2]2] (n [0]2's) has growth rate theta(W^n)

[[[[1]2]2]2] has growth rate SVO

[[[[n]2]2]2] has growth rate theta(W^w^n)

[[[[[0]]2]2]2] has growth rate theta(W^w^w)

[[[[[[0]2]]2]2]2] has growth rate theta(W^e(0))

[[[[[[[0]2]2]]2]2]2] has growth rate theta(W^gamma(0))

[[[[[[[[0]2]2]2]]2]2]2] has growth rate theta(W^SVO)

[[[[[0]2]2]2]2] has growth rate LVO

[[[[[1]2]2]2]2] has growth rate theta(W^W^w)

[[[[[[0]2]2]2]2]2] has growth rate theta(W^W^W)

[[...[0]2]...]2] (n nested) has growth rate theta(W^^(n-2))

[[0]3] has growth rate BHO, because f([0]@+1) = f([...f([0]@)...]@).

[[0]3[0]3] has growth rate theta(e(W+1),1)

[[1]3] has growth rate theta(e(W+1),w)

[[[[0]2]]3] has growth rate theta(e(W+1),w)

[[[[0]3]]3] has growth rate theta(e(W+1),BHO)

[[[0]2]3] has growth rate theta(e(W+1)+1)

[[[[0]2]2]3] has growth rate theta(e(W+1)+W)

[[[[[0]2]2]2]3] has growth rate theta(e(W+1)+W^W)

[[[[0]3]2]3] has growth rate theta(e(W+1)*2)

[[[0]3]3] has growth rate theta(z(W+1))

[[[1]3]3] has growth rate theta(phi(w,W+1))

[[[[0]2]3]3] has growth rate theta(phi(W,1))

[[[[0]3]3]3] has growth rate theta(W_2)

[[[[[0]3]3]3]3] has growth rate theta(W_2^W_2)

[[0]4] has growth rate theta(e(W_2+1))

[[[0]4]4] has growth rate theta(z(W_2+1))

[[[[0]4]4]4] has growth rate theta(W_3)

[[0]n] has growth rate theta(e(W_{n-2}+1))

[[0][0]] has growth rate theta(W_w)

[[0]1[0]] has growth rate TBF

[[0]n[0]] has growth rate theta(e(W_{w+n-1}+1))

[[0][0][0]] has growth rate theta(W_{w*2})

[[0][0]...[0]] (n [0]'s) has growth rate theta(W_{w*n})

[[0][1]] has growth rate theta(W_w^2)

[[0][n]] has growth rate theta(W_w^{n+1})

[[0][[0]]] has growth rate theta(W_w^w)

[[0][[0]2]] has growth rate theta(W_e(0))

[[0][[0]3]] has growth rate theta(W_BHO)

[[0][[0]n]] has growth rate theta(W_theta(e(W_{n-2}+1)))

[[0][[0][0]]] has growth rate theta(W_theta(W_w))

[[0][...[0][0]...]] (n subscripts) has growth rate theta(W_...theta(W_w)...) (n W's)

[[0][0]2] has growth rate theta(W_W)

[[0][0]2...[0]2] (n [0]2's) has growth rate theta(W_{W*n})

[[0][1]2] has growth rate theta(W_{W*w})

[[0][[0]2]2] has growth rate theta(W_W^2)

[[0][0]3] has growth rate theta(W_e(W+1))

[[0][0][0]] has growth rate theta(W_W_w)

[[0]...[0]] (n [0]'s) has growth rate theta(W_..._w) (n-1 W's)

Linear Array Notation

[[[0],1]] has growth rate psi(psi_I(0))

[[[0]...[0],1]] (n [0]'s) has growth rate psi(psi_I(n-1))

[[[1],1]] has growth rate psi(psi_I(w))

[[[0]2,1]] has growth rate psi(psi_I(W))

[[[0]3,1]] has growth rate psi(psi_I(e(W+1)))

[[[0][0],1]] has growth rate psi(psi_I(W_w))

[[[0][0][0],1]] has growth rate psi(psi_I(W_W_w))

[[[[0],1],1]] has growth rate psi(psi_I(psi_I(0)))

[[[0,1]2,1]] has growth rate psi(psi_I(I))

[[[[0],1][0,1]2,1]] has growth rate psi(psi_I(I+psi_I(0)))

[[[0,1]2[0,1]2,1]] has growth rate psi(psi_I(I*2))

[[[0,1]2...[0,1]2,1]] (n [0,1]2's) has growth rate psi(psi_I(I*n))

[[[1,1]2,1]] has growth rate psi(psi_I(I*w))

[[[[0]2,1]2,1]] has growth rate psi(psi_I(I*W))

[[[[0][0],1]2,1]] has growth rate psi(psi_I(I*W_w))

[[[[0],1],1]2,1]] has growth rate psi(psi_I(I*psi_I(0)))

[[[0,1]2,1]2,1]] has growth rate psi(psi_I(I^2))

[[[[0,1]2,1]2,1]2,1]] has growth rate psi(psi_I(I^I))

[[[0,1]3,1]] has growth rate psi(psi_I(e(I+1)))

[[[0,1]2[0,1]3,1]] has growth rate psi(psi_I(e(I+1)+I))

[[[[0,1]3,1]2[0,1]3,1]] has growth rate psi(psi_I(e(I+1)*2))

[[[0,1]3[0,1]3,1]] has growth rate psi(psi_I(e(I+2)))

[[[1,1]3,1]] has growth rate psi(psi_I(e(I+w)))

[[[[0]2,1]3,1]] has growth rate psi(psi_I(e(I+W)))

[[[[0,1]2,1]3,1]] has growth rate psi(psi_I(e(I*2)))

[[[[[0]2,1]2,1]3,1]] has growth rate psi(psi_I(e(I*W)))

[[[[[0,1]2,1]2,1]3,1]] has growth rate psi(psi_I(e(I^2)))

[[[[[0,1]3,1]2,1]3,1]] has growth rate psi(psi_I(e(e(I+1))))

[[[[0,1]3,1]3,1]] has growth rate psi(psi_I(z(I+1))) or psi(psi_I(theta_{I+1}(2)))

[[[[0,1]3[0,1]3,1]3,1]] has growth rate psi(psi_I(eta(I+1))) or psi(psi_I(theta_{I+1}(3)))

[[[[1,1]3,1]3,1]] has growth rate psi(psi_I(phi(w,I+1))) or psi(psi_I(theta_{I+1}(w)))

[[[[[0]2,1]3,1]3,1]] has growth rate psi(psi_I(phi(W,I+1))) or psi(psi_I(theta_{I+1}(W)))

[[[[[0][0],1]3,1]3,1]] has growth rate psi(psi_I(phi(W,I+1))) or psi(psi_I(theta_{I+1}(W_w)))

[[[[[[0],1],1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(psi_I(0))))

[[[[[[0,1]2,1],1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(psi_I(I))))

[[[[[[0,1]3,1],1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(psi_I(e(I+1)))))

[[[[[0,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(I)))

[[[[[[0,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(e(I+1))))

[[[[[[[0,1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(e(I+1))))

[[[[[[[1,1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(theta_{I+1}(w))))

[[[[[[[[0]2,1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(theta_{I+1}(W))))

[[[[[[[[[0],1],1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(theta_{I+1}(psi_I(0)))))

[[[[[[[[0,1]2,1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(theta_{I+1}(I))))

[[[[[[[[[[[0,1]2,1]3,1]3,1]2,1]3,1]3,1]2,1]3,1]3,1]] has growth rate psi(psi_I(theta_{I+1}(theta_{I+1}(theta_{I+1}(I)))))

[[[[[0,1]3,1]3,1]3,1]] has growth rate psi(psi_I(W_{I+1}))

[[[[[[0,1]3,1]3,1]3,1]3,1]] has growth rate psi(psi_I(W_{I+1}^W_{I+1}))

[[[0,1]4,1]] has growth rate psi(psi_I(e(W_{I+1})))

[[[0,1]n,1]] has growth rate psi(psi_I(e(W_{I+n-3})))

[[[0,1][0],1]] has growth rate psi(psi_I(W_{I+w}))

[[[0,1][0]2,1]] has growth rate psi(psi_I(W_{I+W}))

[[[0,1][0][0],1]] has growth rate psi(psi_I(W_{I+W_w}))

[[[0,1][[0],1],1]] has growth rate psi(psi_I(W_{I+psi_I(0)}))

[[[0,1][[0,1]2,1],1]] has growth rate psi(psi_I(W_{I+psi_I(I)}))

[[[0,1][[0,1]n,1],1]] has growth rate psi(psi_I(W_{I+psi_I(e(W_{I+n-3}))}))

[[[0,1][[0,1][0],1],1]] has growth rate psi(psi_I(W_{I+psi_I(W_{I+w})}))

[[[0,1][[0,1][[0],1],1],1]] has growth rate psi(psi_I(W_{I+psi_I(W_{I+psi_I(0)})}))

[[[0,1][[0,1][[0,1][0],1],1],1]] has growth rate psi(psi_I(W_{I+psi_I(W_{I+psi_I(W_{I+w})})}))

[[[0,1][[0,1][[0,1][[0],1],1],1],1]] has growth rate psi(psi_I(W_{I+psi_I(W_{I+psi_I(W_{I+psi_I(0)})})}))

[[[0,1][0,1]2,1]] has growth rate psi(psi_I(W_{I*2}))

[[[0,1][0,1]2...[0,1]2,1]] (n [0,1]2's) has growth rate psi(psi_I(W_{I*(n+1)}))

[[[0,1][n,1]2,1]] has growth rate psi(psi_I(W_{I*w^n}))

[[[0,1][[0,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^2}))

[[[0,1][0,1]2[[0,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^2+I}))

[[[0,1][[0,1]2,1]2[[0,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^2*2}))

[[[0,1][1[0,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^2*w}))

[[[0,1][[0,1]2[0,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^3}))

[[[0,1][[1,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^w}))

[[[0,1][[[0,1]2,1]2,1]2,1]] has growth rate psi(psi_I(W_{I^I}))

[[[0,1][[0,1]3,1]2,1]] has growth rate psi(psi_I(W_e(I+1)))

[[[0,1][[[0,1]3,1]3,1]2,1]] has growth rate psi(psi_I(W_theta_{I+1}(2)))

[[[0,1][[[[0,1]2,1]3,1]3,1]2,1]] has growth rate psi(psi_I(W_theta_{I+1}(I)))

[[[0,1][[[[0,1]3,1]3,1]3,1]2,1]] has growth rate psi(psi_I(W_theta_{I+1}(W_{I+1})))

[[[0,1][[0,1]4,1]2,1]] has growth rate psi(psi_I(W_theta_{I+1}(e(W_{I+1}+1))))

[[[0,1][[0,1][0],1]2,1]] has growth rate psi(psi_I(W_theta_{I+1}(W_{I+w})))

[[[0,1][0,1]3,1]] has growth rate psi(psi_I(W_W_{I+1}))

[[[0,1][0,1][0],1]] has growth rate psi(psi_I(W_W_{I+w}))

[[[0,1][0,1][[0],1],1]] has growth rate psi(psi_I(W_W_{I+psi_I(0)}))

[[[0,1][0,1][0,1]2,1]] has growth rate psi(psi_I(W_W_{I*2}))

[[[0],2]] has growth rate psi(psi_{I_2}(0))

[[[0,1]2,2]] has growth rate psi(psi_{I_2}(I))

[[[0,2]2,2]] has growth rate psi(psi_{I_2}(I_2))

[[[0,2]2[0,2]2,2]] has growth rate psi(psi_{I_2}(I_2*2))

[[[1,2]2,2]] has growth rate psi(psi_{I_2}(I_2*w))

[[[[0]2,2]2,2]] has growth rate psi(psi_{I_2}(I_2*W))

[[[[0,1]2,2]2,2]] has growth rate psi(psi_{I_2}(I_2*I))

[[[[0,2]2,2]2,2]] has growth rate psi(psi_{I_2}(I_2^2))

[[[0,2]3,2]] has growth rate psi(psi_{I_2}(e(I_2+1)))

[[[0,2][0],2]] has growth rate psi(psi_{I_2}(W_{I_2+w}))

[[[0,2][0]2,2]] has growth rate psi(psi_{I_2}(W_{I_2+W}))

[[[0,2][0,1]2,2]] has growth rate psi(psi_{I_2}(W_{I_2+I}))

[[[0,2][0,2]2,2]] has growth rate psi(psi_{I_2}(W_{I_2*2}))

[[[0,2][0,2][0],2]] has growth rate psi(psi_{I_2}(W_W_{I_2+w}))

[[[0,2][0,2][0,1]2,2]] has growth rate psi(psi_{I_2}(W_W_{I_2+I}))

[[[0,2][0,2][0,2]2,2]] has growth rate psi(psi_{I_2}(W_W_{I_2*2}))

[[[0],3]] has growth rate psi(psi_{I_3}(0))

[[[0],n]] has growth rate psi(psi_{I_n}(0))

[[[0],[0]]] has growth rate psi(psi_{I_w}(0))

[[[0],[0]2]] has growth rate psi(psi_{I_W}(0))

[[[0],[0,1]2]]] has growth rate psi(psi_{I_I}(0))

[[[0],[[0],[0]]]] has growth rate psi(psi_{I_I_w}(0))

[[[0],[0,1]2]] has growth rate psi(psi_{I(1,0)}(0))

[[[0],[0,1]2[0,1]2]] has growth rate psi(psi_{I(1,0)}(1))

[[[0],[1,1]2]] has growth rate psi(psi_{I(1,0)}(w))

[[[0],[[[0],1],1]2]] has growth rate psi(psi_{I(1,0)}(psi_I(0)))

[[[0],[[0],[0]]2]] has growth rate psi(psi_{I(1,0)}(I_w))

[[[0],[[0],[0,1]2]2]] has growth rate psi(psi_{I(1,0)}(I(1,0)))

I don't know of limit growth rate of linear array notation. I predict psi(psi_{Xi(w,0)}(0)).

So, can you convert Dollar Function to FGH?

Dimensional Arrays

[[[0](1)1]] has growth rate psi(psi_{Xi(w,0)}(0))

[[[[0](1)1](1)1]] has growth rate psi(psi_{Xi(Xi(w,0),0)}(0))

[[[0(1)1]2(1)1]] has growth rate psi(psi_{Xi(K,0)}(0))

[[[0(1)1]2[0(1)1]2(1)1]] has growth rate psi(psi_{Xi(K*2,0)}(0))

[[[0(1)1]3(1)1]] has growth rate psi(psi_{Xi(e(K+1),0)}(0))

[[[0],1(1)1]] has growth rate psi(psi_{W(W{2}^3+W{2})}(0))

[[[0](1)2]] has growth rate psi(psi_{W(W{2}^3*2)}(0))

[[[0](1)0,1]] has growth rate psi(psi_{W(W{2}^4)}(0))

[[[0](1)0(1)1]] has growth rate psi(psi_{W(W{2}^w)}(0))

[[[0](2)1]] has growth rate psi(psi_{W(W{2}^W{2}^w)}(0))

[[[0]([0])1]] has growth rate psi(psi_{W(W{2}^W{2}^W{2}^w)}(0))

Nested Array Notation

[[[0](0,1)1]] has growth rate psi(psi_{W(e(W{2}+1))}(0))

Hyper-Nested Array Notation

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