[math-fun] Sum and difference of two Roth-type transcendentals are both Shallit-type

15 Aug 2019

      Shallit must know this, but I'm a poor scholar.  Explosive "partial
floors", even case:
In[278]:= ContinuedFraction[Sum[ 2^-2^(2 n), {n, 0, 11}], 32]

Out[278]= {0, 1, 1, 3, 2, 255, 1, 1, 3, 1, 1, 4294967295, 2, 3, 1, 1,
255, 2, 3, 1, 1, 340282366920938463463374607431768211455, 2, 3, 2,
255, 1, 1, 3, 2, 4294967295, 1}

Odd case:
In[279]:= ContinuedFraction[Sum[2^-2^(2 n + 1), {n, 0, 11}], 32]

Out[279]= {0, 3, 1, 15, 4, 65535, 1, 3, 15, 1, 3,
18446744073709551615, 1, 2, 1, 15, 3, 1, 65535, 4, 15, 1, 3,
115792089237316195423570985008687907853269984665640564039457584007913129639935,
1, 2, 1, 15, 4, 65535, 1, 3}

Difference and sum have bounded, predictable partial floors:
In[280]:= ContinuedFraction /@ {#1 - #2, #1 + #2} & @@FromContinuedFraction
/@ {%%, %}

Out[280]= {
 {0, 3, 4, 6, 4, 2, 6, 4, 2, 4, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 4, 2, 6, 4,
4, 2, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4,
  2, 4, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 2, 4, 6, 4,
4, 2, 4, 6, 4, 2, 6, 4, 2, 4, 4, 6, 2, 4, 6, 4,
  4, 2, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 4, 2, 6, 4,
2, 4, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4,

  5, 1, 1, 2, 2, 33, 1, 2, 1, 1, 1, 4, 12, 292, 1, 1, 86, 1, 53, 5, 1, 2,
4, 5, 1, 6, 1, 1, 26, 23, 1, 3, 1, 16, 1, 2, 5, 3, 1, 4, 1, 2, 1, 2, 2, 1,
2, 3, 1, 17, 4, 2, 7,
  6, 1, 5, 1, 29, 1, 8, 1, 1, 2, 1, 1, 2, 5, 1, 1, 3, 3, 1, 3, 2, 1, 21, 1,
1, 2, 1, 1, 1, 19, 3, 6, 1, 3, 2, 2, 2, 2, 62, 3, 1, 13, 1,  49, 1, 1, 3,
2, 5, 5, 6, 7, 4, 7, 1,
  6, 8, 2, 1, 1, 1, 1, 1, 1, 196,3, 1, 2, 2, 1, 3, 1, 74, 832, 1, 29, 1, 5,
3472, 8, 1, 3, 3, 25, 1, 14, 1, 13, 4, 4, 6, 12, 1, 1, 6, 1, 2, 1, 2, 6, 1,
4, 10, 1, 1, 6, 3,
 1, 4, 13, 1, 3, 1, 1, 1, 2, 2, 1, 5, 1, 1, 65, 6, 63, 2, 2, 1, 1, 3, 1, 2,
2, 1, 1, 4, 2, 37, 1, 1, 15, 1, 1, 3, 1, 1, 3, 4, 1, 3, 1,1, 2, 4, 7, 2, 4,
7, 1, 3, 1, 1, 2, 2, 2,
  2, 2, 1, 15, 3, 5, 16, 1, 1, 1, 2, 1, 2, 1, 13},

 {0, 1, 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4,
2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4,
  2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4,
2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4,
  2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4,
2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2,

  5, 1, 1, 1, 2, 6, 1, 1, 14, 2, 1, 3, 7, 2, 1, 291, 1, 8, 1, 8, 1, 8, 1,
11, 3, 20, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 227, 1, 6, 1, 2, 1, 77,
  6, 2, 27, 3, 9, 1, 2, 1, 54, 11, 4, 4, 1, 2, 15, 1, 2, 1, 3, 1, 2, 6, 1,
3, 1, 12,  2, 1, 5, 1, 5, 1, 2, 1, 12, 2, 2, 1, 3, 8, 1, 5, 1, 37,
  9, 1, 1, 3, 156, 2, 26, 3, 1, 4, 2, 3, 1, 3, 116, 1, 14, 1, 5, 1, 5, 4,
3, 1, 5, 1, 1, 3, 5, 8, 1, 2, 1, 1, 1, 7, 1, 1, 1, 9, 4, 3, 2, 2,
 2, 2, 7, 1, 4, 4, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 121, 2, 3, 4, 5, 2, 1, 1,
1, 1, 2, 11, 7, 1, 3, 134, 15, 3, 7, 1, 1, 2, 13, 1, 1, 2, 4, 2,
 2, 1, 3, 1, 2, 1, 5, 2, 2, 2, 6, 1, 3, 16, 2, 1, 1, 24, 3, 7, 8, 1, 1, 6,
2, 1, 1, 11, 4, 2, 1, 8756, 1, 1, 50, 1, 1, 1, 4, 1, 1, 25, 1, 5,
 5,1, 2, 5, 4, 1, 52, 1, 55, 2, 15}}

(Presumably the eventual noise is an artifact of series truncation.)  For
an (almost) explanation of the sum pattern
in ContinuedFraction[Sum[ u^-2^n, {n, 0, ∞}], ...],
https://cs.uwaterloo.ca/~shallit/Papers/scf.ps,
except it only treats integer u>2, presumably because the u=2 pattern is
haired up by vanishing partial floors.

Also:  In[288]:= ContinuedFraction[Sum[2^n 4^-2^n, {n, 0, 13}], 64]

Out[288]= {0, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1,
1, 3, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 7, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1,
1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 15}

In[289]:= ContinuedFraction[Sum[(-2)^n 4^-2^n, {n, 0, 13}], 64]

Out[289]= {0, 7, 8, 1, 1, 9, 7, 3, 1, 6, 9, 1, 1, 8, 6, 1, 7, 7, 8,
1, 1, 9, 6, 1, 3, 7, 9, 1, 1, 8, 7, 15, 1, 6, 8, 1, 1, 9, 7, 3, 1, 6,
9, 1, 1, 8, 7, 7, 1, 6, 8, 1, 1, 9, 6, 1, 3, 7, 9, 1, 1, 8, 6, 1}

(Probably unbounded but predictable.)  Likewise:
In[286]:= ContinuedFraction[Sum[(-2)^n 2^-2^n, {n, 0, 13}], 64]

Out[286]= {0, 4, 1, 1, 3, 3, 1, 2, 1, 1, 3, 1, 7, 4, 1, 1, 2, 1, 3,
3, 1, 1, 4, 15, 1, 3, 1, 1, 3, 3, 1, 2, 1, 1, 4, 7, 1, 3, 1, 1, 2, 1,
3, 3, 1, 1, 3, 1, 31, 4, 1, 1, 3, 3, 1, 2, 1, 1, 3, 1, 7, 4, 1, 1}

In[287]:= ContinuedFraction[Sum[2^n 3^-2^n, {n, 0, 13}], 64]

Out[287]= {0, 1, 1, 1, 1, 5, 1, 8, 1, 1, 2, 1, 1, 15, 1, 1, 1, 2, 6,
1, 1, 1, 10, 31, 1, 10, 4, 3, 1, 9, 1, 2, 1, 2, 1, 1, 3, 5, 1, 14, 1,
1, 1, 63, 2, 1, 14, 1, 5, 6, 2, 3, 1, 7, 4, 2, 1, 1, 3, 1, 3, 10, 4, 3}
—rwg

Bill Gosper

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