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A065444
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Decimal expansion of 9*Sum_{k>=1} 1/(10^k-1).
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6
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1, 1, 0, 0, 9, 1, 8, 1, 9, 0, 8, 3, 6, 2, 0, 0, 7, 3, 6, 3, 7, 9, 8, 5, 5, 1, 0, 1, 6, 5, 4, 3, 8, 0, 0, 4, 3, 2, 0, 3, 4, 5, 4, 3, 9, 7, 8, 7, 3, 2, 8, 1, 6, 5, 6, 3, 5, 9, 8, 9, 0, 2, 2, 0, 7, 3, 4, 3, 8, 3, 4, 9, 0, 2, 1, 9, 8, 3, 4, 7, 4, 8, 8, 9, 2, 0, 0, 3, 4, 9, 2, 1, 8, 0, 0, 7, 0, 4, 0, 2, 3, 5
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OFFSET
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1,5
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COMMENTS
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
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LINKS
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FORMULA
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Equals 9 * sum(k>=1, (1+x^k)/(1-x^k) * x^(k^2) ) where x = 1/10. This allows fast computation for this and similar sequences (involving sum(k>=1, x^k/(1-x^k) for some x < 1 ). - Joerg Arndt, Apr 25 2016
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EXAMPLE
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1.10091819083620073637985510165438004320345439787328165635989...
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MATHEMATICA
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RealDigits[9*N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]
A065444=RealDigits[ Block[{$MaxExtraPrecision = 100}, N[9*Sum[(-1 + 10^i)^-1, {i, 1, Infinity}], 130]]][[1]] (* Enrique Pérez Herrero, Dec 06 2009 *)
First[RealDigits[9 (Log10[10/9] - QPolyGamma[0, 1, 1/10]/Log[10]), 10, 120]] (* Jan Mangaldan, Apr 25 2016 *)
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PROG
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(PARI) { default(realprecision, 2080); x=9*suminf(k=1, 1/(10^k - 1)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065444.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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...733 (50th digit) expanded to ...7328165 etc. by Frank Ellermann, Feb 23 2002
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STATUS
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approved
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