OFFSET
0,1
COMMENTS
This constant plus A293382 equals log(2), due to the identity:
Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 1/2, y = 1/3.
FORMULA
Constant: Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).
Constant: log(2) - Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).
EXAMPLE
Constant t = 0.3000496898598647328718775015850557230541585593534995437986897...
such that
t = (3 - 2)/(1*2*3) + (3^2 - 2)^2/(2*2^2*3^4) + (3^3 - 2)^3/(3*2^3*3^9) + (3^4 - 2)^4/(4*2^4*3^16) + (3^5 - 2)^5/(5*2^5*3^25) + (3^6 - 2)^6/(6*2^6*3^36) + (3^7 - 2)^7/(7*2^7*3^49) +...+ (3^n - 2)^n / (n * 2^n * 3^(n^2)) +...
More explicitly,
t = 1/(1*2*3) + 7^2/(2*4*3^4) + 25^3/(3*8*3^9) + 79^4/(4*16*3^16) + 241^5/(5*32*3^25) + 727^6/(6*64*3^36) + 2185^7/(7*128*3^49) + 6559^8/(8*256*3^64) + 19681^9/(9*512*3^81) + 59047^10/(10*1024*3^100) + 177145^11/(11*2048*3^121) + 531439^12/(12*4096*3^144) +...
Also,
log(2) - t = 2/(2*3-1) - 2^2/(2*(2*3^2-1)^2) + 2^3/(3*(2*3^3-1)^3) - 2^4/(4*(2*3^4-1)^4) + 2^5/(5*(2*3^5-1)^5) - 2^6/(6*(2*3^6-1)^6) + 2^7/(7*(2*3^7-1)^7) - 2^8/(8*(2*3^8-1)^8) +...+ -(-1)^n * 2^n / (n * (2*3^n - 1)^n) +...
PROG
(PARI) {t = suminf(n=1, 1.*(3^n - 2)^n / (n * 2^n * 3^(n^2)) )}
for(n=1, 120, print1(floor(10^n*t)%10, ", "))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, Oct 12 2017
STATUS
approved