[seqfan] Binary Expansion of Constant A303340

Paul Hanna pauldhanna.math at gmail.com
Wed Apr 25 00:21:50 CEST 2018


SeqFans,
      Can someone find a nice construction for the binary expansion of the
constant defined by https://oeis.org/A303340 ?

This constant equals the sum of the following infinite series.

(1) B = 1 - 1/3^2 + 1/7^3 - 1/15^4 + 1/31^5 - 1/63^6 + 1/127^7 - 1/255^8 +
1/511^9 - 1/1023^10 + 1/2047^11 - 1/4095^12 + 1/8191^13 - 1/16383^14 + ...
+  (-1)^(n-1) / (2^n - 1)^n + ...

(2) B = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 +
127^6/2^49 + 255^7/2^64 + 511^8/2^81 + 1023^9/2^100 + 2047^10/2^121 +
4095^11/2^144 + ... +  (2^n - 1)^(n-1) / 2^(n^2) + ...

In base 10: B = 0.891784622610953349715890136060239421022216970366139...

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.

Thanks,
       Paul



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