[seqfan] Re: Binary Expansion of Constant A303340

[M. F. Hasler]

On Tue, Apr 24, 2018 at 6:21 PM, Paul Hanna <pauldhanna.math at gmail.com> wrote:
> SeqFans,
>       Can someone find a nice construction for the binary expansion of the
> constant defined by https://oeis.org/A303340 ?
> (2) B = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 +
> + ... +  (2^n - 1)^(n-1) / 2^(n^2) + ...
> In base  2: B = 0.11100100010010111111111100111101001100001001001100...
>
> From the above series, it seems like the binary digits of this constant
> should have some pattern or method of construction.

In B = Sum_n  (2^n - 1)^(n-1) / 2^(n^2)
the length of the numerator is  1, 2, 6, 12, 20, 30, 42, 56, = n*(n-1) for n>1.
Since these numerators are shifted n^2 bits to the right,
there is a large overlap between successive numerators:
n=1: 0.1  (length 1 ; positions 1..1
n=2: 0.0011 (length 2, positions 3..4 : no overlap with preceding term
n=3: 0.000110001 (length 6, positions 4..9 : overlap of the leftmost
digit with the preceding term)
n=4: 0.0000110100101111 (length 12, positions 5..16 : overlap of 5
digits with the preceding term)
n=5: 0.0000011100001011110000001 (length 20, positions 6..20 : overlap
of 11 digits with the preceding term)
...
Although nonzero bits of numerator n only start at position n+1, they
will influence earlier digits of several preceding terms:
e.g., the second nonzero bit in B comes from terms n=2 and n=3, but
the third nonzero bit comes from much later:
the sum from n=3 and n=4 yields a 1 in position 4 (result of the sum
of two bits 1 in position 5 of both),
and only the sum of terms n=5 and n=6 will result in adding 1 to that
bit, which yields the "final" 1 in position 3.

I think it is highly nontrivial to get a closed expression for the
nonzero bits, and one could add the list of indices of these as a
separate OEIS sequence.
Note that the indices of this "cons" sequence are, by current
conventions, 0,-1,-2,-3, ...
In the hope of maybe getting one day a simple formula for this, I
think it would be much more sensible to index the terms 1,2,3,...;
 in that case, the index would be the power of 2 which corresponds to
the given digit.
That sequence would start
1,2,3,6,10,13,15,16,17,18,19,20,21,22,23,24,27,28,29,30,32,35,36,41,44,47,48,51,52,53,54,55,57,62,65,66,68,69,72,74,81,82,86,87,88,...

[As a side note, irrelevant for this problem, [[Warning: politically
incorrect "free speech" following - please ignore!]]
I've been suggesting for many years that "cons" sequences should
generally be indexed in this very logical way, as to have:
* const = Sum_{i >= offset} a(i)*b^(-i),
* increasing indices for all sequences,
* at most a finite number of "ugly" negative terms
* no contradiction between indices in the "list" and in the b-file,
* none of all the other problems arising from the IMHO weird current
index convention for "cons" sequences, which will certainly change
sooner or (possibly much) later.]

One could also add the sequence of run lengths of 0's and 1's
(independent of the indexing scheme):
3,2,1,3,1,2,1,1,10,2,4,1,1,2,2,4,1,2,1,2,2,2,5,1,1,4,1,2,2,1,2,2,1,1,1,6,2,3,3,1,1,3,1,2,4,1,1,1,1,1,1,1,1,3,1,6,6,4,1,1,2,1,4,1,1,4,....
Neither for this nor for the run lengths of 1's or of 0's,
3,1,1,1,10,4,1,2,1,1,2,5,1,1,2,2,1,1,2,3,1,1,4,1,1,1,1,1,6,1,2,4,1,6,3,1,2,1,1,1,1,3,2,2,1,1,1,1,1,2,2,1,1,1,2,1,3,2,2,1,1,3,6,3,5,1...
respectively for 0's:
2,3,2,1,2,1,2,4,2,2,2,1,4,2,1,2,1,6,3,1,3,2,1,1,1,1,3,6,4,1,1,1,4,4,2,1,1,3,1,1,1,1,4,2,4,1,1,4,2,1,2,1,1,1,3,2,1,3,1,1,4,1,1,1,3,1,...
I or OEIS could find something.

Cheers,
Maximilian