Thus there are only three diminished seventh chords before the pattern repeats.
Which makes for interesting possibilities for key modulation. I mention this because I found it very cool when someone told me about it a long time ago.
Since Adim7 = Cdim7 = Ebdim7 = Gbdim7 (i.e., these are all the same chords), having landed on any inversion, you stand in the same harmonic relation to four different keys.
So, if you start in C major and find your way to a Cdim7 chord, you can say, OK, now I'm in A major, and take it from there. Then you can find your way to an Adim7 chord and switch right back to C major – or Eb, or Gb.
I recorded this 28-second thing back in 1981 using a Casio VL-Tone, to demonstrate!
https://soundcloud.com/mark_hayes/diminished-modulations