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

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Oct 05 2017

Keywords

• nonn
• cons

Comments

This constant plus A292178 equals log(2), due to the identity (at x = 1/2):

Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / n = -log(1-x).

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

Examples

			Constant t = 0.06651104177050896969800800417721390883141679592591835385754710...
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) +...
Also,
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)
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A292178.

Formula

Constant: Sum_{n>=1} (2^(n-1) - 1)^n / (n * 2^(n^2)).

Constant: log(2) - Sum_{n>=1} -(-1)^n * 2^n / (n * (2^(n+1) - 1)^n).