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A293384
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Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).
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3
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7, 0, 5, 8, 4, 0, 7, 1, 3, 1, 1, 3, 0, 4, 2, 1, 7, 9, 5, 3, 6, 1, 3, 3, 3, 5, 9, 6, 6, 1, 2, 9, 7, 6, 1, 3, 3, 0, 4, 7, 6, 7, 1, 0, 8, 7, 8, 0, 4, 8, 5, 9, 6, 5, 5, 3, 2, 3, 0, 5, 9, 8, 9, 7, 6, 4, 3, 1, 9, 0, 9, 5, 2, 2, 1, 5, 1, 1, 3, 7, 5, 0, 2, 3, 9, 0, 8, 5, 3, 6, 6, 5, 0, 2, 5, 5, 8, 4, 3, 2, 7, 4, 7, 5, 1, 0, 6, 2, 0, 5, 2, 4, 3, 3, 0, 0, 3, 0, 7, 8
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OFFSET
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0,1
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COMMENTS
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This constant plus A293383 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.
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LINKS
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FORMULA
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Constant: Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).
Constant: log(3) - Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).
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EXAMPLE
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Constant t = 0.7058407131130421795361333596612976133047671087804859655323059...
such that
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) - 3^8/(8*256*(3*2^7-1)^8) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...
More explicitly,
t = 3/(1*2*2) - 9/(2*4*5^2) + 27/(3*8*11^3) - 81/(4*16*23^4) + 243/(5*32*47^5) - 729/(6*64*95^6) + 2187/(7*128*191^7) - 6561/(8*256*383^8) + 19683/(9*512*767^9) - 59049/(10*1024*1535^10) + 177147/(11*2048*3071^11) - 531441/(12*4096*6143^12) +...
Also,
log(3) - 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)) +...
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PROG
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(PARI) {t = suminf(n=1, -1.*(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n) )}
for(n=1, 120, print1(floor(10^n*t)%10, ", "))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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