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Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).
3

%I #6 Oct 14 2017 05:50:18

%S 3,9,2,7,7,1,5,7,5,5,5,5,0,6,7,5,1,1,8,5,9,1,1,1,8,7,7,2,6,1,2,2,8,0,

%T 9,1,3,4,2,7,2,3,4,4,9,0,4,2,2,6,3,4,8,6,2,0,2,3,8,8,3,4,3,8,7,3,1,7,

%U 5,1,9,7,9,9,7,0,9,7,5,9,1,8,4,9,7,0,7,2,1,8,1,6,3,4,7,6,2,4,5,5,1,3,2,1,8,9,6,7,0,1,3,5,2,4,8,6,2,6,6,3

%N Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

%C This constant plus A293384 equals log(3), due to the identity:

%C Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

%F Constant: Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

%F Constant: log(3) - Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

%e Constant t = 0.3927715755550675118591118772612280913427234490422634862023883438....

%e such that

%e t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...

%e More explicitly,

%e t = 1/(1*3*2) + 5^2/(2*9*2^4) + 13^3/(3*27*2^9) + 29^4/(4*81*2^16) + 61^5/(5*243*2^25) + 125^6/(6*729*2^36) + 253^7/(7*2187*2^49) + 509^8/(8*6561*2^64) + 1021^9/(9*19683*2^81) + 2045^10/(10*59049*2^100) + 4093^11/(11*177147*2^121) + 8189^12/(12*531441*2^144) +...

%e Also,

%e log(3) - t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...

%o (PARI) {t = suminf(n=1, 1.*(2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)) )}

%o for(n=1,120, print1(floor(10^n*t)%10,", "))

%Y Cf. A293384, A293381, A293382, A292178, A292179, A002391 (log 3).

%K nonn,cons

%O 0,1

%A _Paul D. Hanna_, Oct 13 2017