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%I #20 Oct 14 2017 05:49:11
%S 3,9,3,0,9,7,4,9,0,7,0,0,0,8,0,5,7,6,5,4,5,3,5,4,6,1,9,8,7,3,1,2,0,8,
%T 4,5,0,2,1,3,4,1,5,7,5,0,0,6,7,5,5,7,1,0,3,2,1,9,9,0,3,0,8,0,3,2,4,7,
%U 8,8,6,7,5,3,5,7,0,7,5,7,7,4,9,8,8,6,6,3,5,5,7,6,2,2,2,4,2,3,6,9,9,7,9,5,6,4,8,7,5,4,4,9,9,3,7,8,5,0,5,9
%N Decimal expansion of Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).
%C This constant plus A293381 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} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).
%F Constant: log(2) - Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).
%e Constant t = 0.3930974907000805765453546198731208450213415750067557103219903...
%e such that
%e 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) +...
%e More explicitly,
%e t = 2/5 - 4/(2*17^2) + 8/(3*53^3) - 16/(4*161^4) + 32/(5*485^5) - 64/(6*1457^6) + 128/(7*4373^7) - 256/(8*13121^8) + 512/(9*39365^9) - 1024/(10*118097^10) +...
%e Also,
%e log(2) - 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)) +...
%o (PARI) {t = suminf(n=1, -1.*(-1)^n * 2^n / (n * (2*3^n - 1)^n) )}
%o for(n=1,120, print1(floor(10^n*t)%10,", "))
%Y Cf. A002162, A293381, A292178, A292179, A293383, A293384.
%K nonn,cons
%O 0,1
%A _Paul D. Hanna_, Oct 13 2017
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