Evaluating the Svein-Graham Sum

[6c 6f 76 65]

Good evening dearest physicians and mathematicians,

I recently came across the so-called "Svein-Graham sum", and i wondered: is it possible to find a simple formula for evaluating it?
$\sum_{i=0}^k x\uparrow\uparrow i = \left .1+x+x^x+x^{x^x}+ ... +x^{x^{x^{x^{.^{.^{.^x}}}}}}\right \}k$

 

[Quantioner]

Hi, I use Mathematica to define a function sg[x,k] to calculate the Svein-Graham sum and plot some figures for $x \in [1,2]$ with $k$ varies from 1 to 5.

sg[x_, k_] := Module[{f},
  f[y_] := #^y &;
  (FoldList[f[x], x, Range[k - 1]] // Total) + 1]
Plot[sg[x, #], {x, 1, 2}] & /@ Range[1, 5]

 

[6c 6f 76 65]

Hi, I use Mathematica to define a function sg[x,k] to calculate the Svein-Graham sum and plot some figures for $x \in [1,2]$ with $k$ varies from 1 to 5.

sg[x_, k_] := Module[{f},
  f[y_] := #^y &;
  (FoldList[f[x], x, Range[k - 1]] // Total) + 1]
Plot[sg[x, #], {x, 1, 2}] & /@ Range[1, 5]

I was looking for a more analytic expression like $\sum_{i=1}^n i = \frac{n(n+1)}{2}$. Maybe it's possible to find yet another connection to the Bernoulli numbers? But thank you nevertheless!