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%I #21 Oct 14 2017 05:48:20
%S 3,0,0,0,4,9,6,8,9,8,5,9,8,6,4,7,3,2,8,7,1,8,7,7,5,0,1,5,8,5,0,5,5,7,
%T 2,3,0,5,4,1,5,8,5,5,9,3,5,3,4,9,9,5,4,3,7,9,8,6,8,9,7,0,1,4,6,0,9,1,
%U 4,7,5,4,4,3,3,9,8,7,1,3,8,1,0,6,9,9,6,9,7,1,2,3,4,1,9,4,4,5,0,5,4,4,0,4,4,9,9,3,4,7,5,5,7,6,9,0,0,6,7,4
%N Decimal expansion of Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).
%C This constant plus A293382 equals log(2), due to the identity:
%C Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 1/2, y = 1/3.
%F Constant: Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).
%F Constant: log(2) - Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).
%e Constant t = 0.3000496898598647328718775015850557230541585593534995437986897...
%e such that
%e 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)) +...
%e More explicitly,
%e 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) +...
%e Also,
%e 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) +...
%o (PARI) {t = suminf(n=1, 1.*(3^n - 2)^n / (n * 2^n * 3^(n^2)) )}
%o for(n=1,120, print1(floor(10^n*t)%10,", "))
%Y Cf. A002162, A293382, A292178, A292179, A293383, A293384.
%K nonn,cons
%O 0,1
%A _Paul D. Hanna_, Oct 12 2017