cp's OEIS Frontend

A292179 Decimal expansion of: Sum_{n>=1} (1/2 - 1/2^n)^n / n.

%I A292179 #20 Mar 03 2025 13:32:17
%S A292179 0,6,6,5,1,1,0,4,1,7,7,0,5,0,8,9,6,9,6,9,8,0,0,8,0,0,4,1,7,7,2,1,3,9,
%T A292179 0,8,8,3,1,4,1,6,7,9,5,9,2,5,9,1,8,3,5,3,8,5,7,5,4,7,1,0,3,2,4,4,1,6,
%U A292179 3,5,1,0,2,8,8,2,0,5,9,6,7,2,1,0,7,1,9,3,5,7,4,5,0,5,2,0,9,6,3,7,3,2,9,0,1,7,0,3,6,5,2,0,8,7,7,3,4,6,4,8,9,6,8,2,6,9,7,8,6,3,2,0,3,8,7,0,2,2,1,4,8,7,1,5,1,7,7,9,6,0
%N A292179 Decimal expansion of: Sum_{n>=1} (1/2 - 1/2^n)^n / n.
%C A292179 This constant plus A292178 equals log(2), due to the identity (at x = 1/2):
%C A292179 Sum_{n=-oo..+oo, n<>0}  (x - x^n)^n / n  =  -log(1-x).
%C A292179 Conjecture: Sum_{n in Z - {0}} (x - a^n)^n / n = -log(1 - x) for |x| < 1 and |a| < 1. - _Peter Bala_, Mar 02 2025
%H A292179 Paul D. Hanna, <a href="/A292179/b292179.txt">Table of n, a(n) for n = 0..500</a>
%F A292179 Constant: Sum_{n>=1} (2^(n-1) - 1)^n / (n * 2^(n^2)).
%F A292179 Constant: log(2) - Sum_{n>=1}  -(-1)^n * 2^n / (n * (2^(n+1) - 1)^n).
%e A292179 Constant t = 0.06651104177050896969800800417721390883141679592591835385754710...
%e A292179 where t = 0/(1*2) + 1^2/(2*2^4) + 3^3/(3*2^9) + 7^4/(4*2^16) + 15^5/(5*2^25) + 31^6/(6*2^36) + 63^7/(7*2^49) + 127^8/(8*2^64) + 255^9/(9*2^81) + 511^10/(10*2^100) + 1023^11/(11*2^121) + 2047^12/(12*2^144) + 4095^13/(13*2^169) + 8191^14/(14*2^196) + 16383^15/(15*2^225) +...
%e A292179 Also,
%e A292179 log(2) - t = 2/(1*3) - 4/(2*7^2) + 8/(3*15^3) - 16/(4*31^4) + 32/(5*63^5) - 64/(6*127^6) + 128/(7*255^7) - 256/(8*511^8) + 512/(9*1023^9) - 1024/(10*2047^10) + 2048/(11*4095^11) - 4096/(12*8191^12) + 8192/(13*16383^13) - 16384/(14*32767^14) + 32768/(15*65535^15) +... (constant A292178)
%Y A292179 Cf. A292178.
%K A292179 nonn,cons
%O A292179 0,2
%A A292179 _Paul D. Hanna_, Oct 05 2017