A293383 Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Oct 13 2017

Keywords

• nonn
• cons

Comments

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

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

Examples

			Constant t = 0.3927715755550675118591118772612280913427234490422634862023883438....
such that
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)) +...
More explicitly,
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) +...
Also,
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) +...
		

Crossrefs

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

Programs

PARI

{t = suminf(n=1, 1.*(2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Formula

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

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