Re: [math-fun] FromContinuedFraction@{0, 1, 1, 2, 2, 3, 3, 4, 4, . . .}

15 Sep 2018


      This is weird.
In[1]:= FromContinuedFraction@Flatten@Transpose@{#, #} &@Range@69

Out[1]= 19195582280919906386643729491698663192728164717800401264661169\
3236922426237014409271540224025631997628114767040193942539601948574614\
694291216848805119377885379972463008056389222247094056500011138035/\
1123320225289260778343857878081309971450448336209009645040260404323103\
9234921896631339460535953076162653478097057716058925988033482585395692\
6778133515773376078911965787578144121657890600700433584701

In[2]:= ContinuedFraction@%

Out[2]= {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, \
10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, \
19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, \
27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, \
36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, \
44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 51, 51, 52, 52, \
53, 53, 54, 54, 55, 55, 56, 56, 57, 57, 58, 58, 59, 59, 60, 60, 61, \
61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 68, 68, 69, 69}

In[3]:= ContinuedFraction[(3 %% - 2)/(2 %% + 1)]

Out[3]= {0, 1, 2, 2, 2, 1, 2, 7, 6, 5, 5, 4, 4, 1, 2, 4, 3, 1, 2, 3, \
4, 6, 1, 6, 3, 3, 1, 6, 2, 1, 2, 2, 6, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, \
1, 1, 2, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, \
2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 2, 3, 2, 2, 6, 1, 1, 1, 2, 1, 1, 2, \
6, 1, 1, 1, 6, 2, 1, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 2, 3, 2, 2, 1, \
2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, \
2, 1, 6, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 1, 1, 2, 1, \
3, 3, 2, 3, 1, 1, 2, 1, 3, 2, 3, 3, 1, 6, 4, 6, 1, 3, 2, 3, 4, 6, 1, \
3, 1, 1, 2, 1, 3, 1, 6, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 2, 1, 4, \
6, 1, 4, 6, 1, 4, 6, 1, 4, 3, 2, 4, 1, 2, 1, 1, 4, 1, 2, 1, 1, 4, 1, \
2, 1, 1, 4, 1, 6, 5, 3, 2, 5, 6, 1, 4, 1, 2, 1, 1, 5, 6, 1, 4, 1, 6, \
5, 1, 2, 1, 1, 5, 3, 2, 5, 1, 1, 2, 1, 5, 3, 2, 5, 1, 2, 1, 1, 5, 1, \
1, 2, 1, 5, 1, 1, 2, 1, 5, 1, 1, 2, 1, 5, 1, 2, 1, 1, 5, 1, 6, 6, 2, \
3, 6, 3, 2, 6, 3, 2, 6, 3, 2, 6, 2, 3, 6, 1, 1, 2, 1, 6, 3, 2, 6, 1, \
1, 2, 1, 6, 2, 3, 6, 1, 6, 7, 6, 1, 6, 2, 3, 7, 6, 1, 6, 1, 1, 2, 1, \
6, 1, 6, 7, 2, 3, 7, 2, 3, 7, 2, 3, 7, 1, 1, 2, 1, 7, 6, 1, 7, 6, 1, \
7, 6, 1, 7, 3, 2, 7, 1, 2, 1, 1, 7, 1, 2, 1, 1, 7, 1, 2, 1, 1, 7, 1, \
6, 8, 3, 2, 8, 6, 1, 7, 1, 2, 1, 1, 8, 6, 1, 7, 1, 6, 8, 1, 2, 1, 1, \
8, 3, 2, 8, 1, 1, 2, 1, 8, 3, 2, 8, 1, 2, 1, 1, 8, 1, 1, 2, 1, 8, 1, \
1, 2, 1, 8, 1, 1, 2, 1, 8, 1, 2, 1, 1, 8, 1, 6, 9, 2, 3, 9, 3, 2, 9, \
3, 2, 9, 3, 2, 9, 2, 3, 9, 1, 1, 3}

In[4]:= ContinuedFraction[(2 %%% - 3)/(%%% + 2)]

Out[4]= {0, 8, 1, 7, 2, 1, 7, 5, 5, 4, 1, 2, 7, 7, 7, 4, 3, 2, 1, 2, \
2, 1, 2, 2, 1, 2, 6, 1, 3, 2, 1, 6, 2, 2, 1, 1, 1, 6, 2, 6, 1, 1, 2, \
1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, \
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 6, 2, 2, 3, 2, 3, \
2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 1, \
2, 2, 3, 2, 1, 6, 3, 6, 1, 2, 2, 3, 3, 6, 1, 2, 1, 1, 2, 1, 2, 1, 6, \
3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 1, 1, 2, 1, 3, 6, 1, 3, 6, 1, 3, 6, 1, \
3, 3, 2, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 3, 1, 6, 4, 3, \
2, 4, 6, 1, 3, 1, 2, 1, 1, 4, 6, 1, 3, 1, 6, 4, 1, 2, 1, 1, 4, 3, 2, \
4, 1, 1, 2, 1, 4, 3, 2, 4, 1, 2, 1, 1, 4, 1, 1, 2, 1, 4, 1, 1, 2, 1, \
4, 1, 1, 2, 1, 4, 1, 2, 1, 1, 4, 1, 6, 5, 2, 3, 5, 3, 2, 5, 3, 2, 5, \
3, 2, 5, 2, 3, 5, 1, 1, 2, 1, 5, 3, 2, 5, 1, 1, 2, 1, 5, 2, 3, 5, 1, \
6, 6, 6, 1, 5, 2, 3, 6, 6, 1, 5, 1, 1, 2, 1, 5, 1, 6, 6, 2, 3, 6, 2, \
3, 6, 2, 3, 6, 1, 1, 2, 1, 6, 6, 1, 6, 6, 1, 6, 6, 1, 6, 3, 2, 6, 1, \
2, 1, 1, 6, 1, 2, 1, 1, 6, 1, 2, 1, 1, 6, 1, 6, 7, 3, 2, 7, 6, 1, 6, \
1, 2, 1, 1, 7, 6, 1, 6, 1, 6, 7, 1, 2, 1, 1, 7, 3, 2, 7, 1, 1, 2, 1, \
7, 3, 2, 7, 1, 2, 1, 1, 7, 1, 1, 2, 1, 7, 1, 1, 2, 1, 7, 1, 1, 2, 1, \
7, 1, 2, 1, 1, 7, 1, 6, 8, 2, 3, 8, 3, 2, 8, 3, 2, 8, 3, 2, 8, 2, 3, \
8, 1, 1, 2, 1, 8, 3, 2, 8, 1, 1, 2, 1, 8, 2, 3, 8, 1, 6, 9, 6, 1, 8, \
2, 3, 9, 6, 1, 8, 1, 1, 2, 1, 8, 1, 6, 9, 2, 3, 9, 2, 3, 9, 2, 3, 9, \
1, 1, 2, 1, 9, 7}
—rwg

On Thu, Aug 9, 2018 at 12:56 PM Bill Gosper <billgosper@gmail.com> wrote:
...
Finch, p424 (Euler-Gompertz Constant):
"No one knows the exact outcome if we instead repeat each denominator
in c1 [ := FromContinuedFraction@{0, 1, 2, 3, 4,. . .} = I_1(2)/I_0(2)] ~
 .6977746579640] although numerically we find that c2 = .5851972651050...
".
Using my mysteriously wrong-but-correctible matrix-to-CF (non-regular)
converter, c2=
1/2 + 1/(12 + 2*ContinuedFractionK[1 - k^2, 1 + 3*k + 2*k^2 + k^3, {k, 2,
∞}]).
Julian & I had a way to further convert these to log-derivatives of  pFqs,
if
anyone is really interested.  --rwg