This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n = 4 there are 5 functions: f1(x) = ((x^x)^x)^x; f2(x) = (x^(x^x))^x; f3(x) = x^((x^x)^x); f4(x) = x^(x^(x^x)); f5(x) = (x^x)^(x^x); but only 4 different values when x = sqrt(2).
trees[1] = {x};
trees[n_] := trees[n] = Flatten@Table[ch1^ch2, {k, n-1}, {ch1, trees[k]}, {ch2, trees[n-k]}];
logs[t_] := ((log/@t) //. {log[a_^b_]:>log[a]b, log[a_ b_]:>log[a]+log[b], log[x]->one, log[one]->0});
Table[Length@Union[logs@logs@trees[n] /. {one->1, x->Sqrt[2]}, SameTest->Equal], {n, 9}] (* Andrei Zabolotskii, Jan 03 2025 *)
a(1) = 1: there is one value, i. a(2) = 1: there is one value, i^i = exp(i Pi / 2)^i = exp(-Pi/2) = 0.2079... a(3) = 2: there are two values: (i^i)^i = i^(-1) = 1/i = -i and i^(i^i) = i^0.2079... = exp(0.2079... i Pi / 2) = 0.9472... + 0.3208... i. a(4) = 3: there are 5 possible parenthesizations but they give only 3 distinct values: i^(i^(i^i)), i^((i^i)^i) = ((i^i)^i)^i, (i^i)^(i^i) = (i^(i^i))^i.
iParens[1] = {I}; iParens[n_] := iParens[n] = Union[Flatten[Table[Outer[Power, iParens[k], iParens[n - k]], {k, n - 1}]], SameTest -> Equal]; Table[Length[iParens[n]], {n, 10}]
a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.
f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0]},
{seq(seq(seq( [2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);
f[n_] := f[n] = If[n == 1, {{0, 0, 0}}, Union @ Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)
a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.
f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0, 0]},
{seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
+15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[4], f(n))):
seq(a(n), n=1..20);
f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length@Union@(#[[4]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)
a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.
f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0, 0, 0]},
{seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
+15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4],
45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+
270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+
60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+
100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[5], f(n))):
seq(a(n), n=1..15);
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