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Search: id:A131917
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| A131917 |
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Decimal expansion of 1 / (1 - gamma - ln(3/2)) - 54, where gamnma is the Euler-Mascheroni constant. |
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+0
4
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| 3, 7, 3, 9, 2, 9, 7, 5, 1, 9, 4, 5, 1, 1, 8, 4, 2, 0, 7, 3, 6, 6, 3, 3, 2, 8, 6, 9, 6, 8, 6, 1, 5, 1, 7, 2, 5, 6, 6, 2, 6, 3, 6, 8, 5, 4, 5, 6, 4, 1, 9, 2, 1, 7, 8, 3, 0, 7, 8, 9, 8, 1, 2, 1, 0, 0, 7, 9, 5, 7, 2, 3, 2, 6, 2, 0, 3, 5, 2, 5, 4, 5, 3, 0, 1, 7, 9, 7, 0, 9, 4, 2, 3, 7, 1, 7, 7, 6, 2, 2, 8, 5, 8, 3, 1
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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Continued fraction expansion is A131918. Abstract: An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the n-th harmonic number into negative powers of the n-th triangular number. We also discuss the history of the Ramanujan expansion for the n-th harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.
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LINKS
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Mark B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number. Constant occurs in Theorem 7 (DeTemple-Wang), formula (1.14), page 6.
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FORMULA
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(54 ln(3/2) + 54 gamma - 53)/(1 - ln(3/2) - gamma) = 1 / (1 - gamma - ln(3/2)) - 54, where Martin Fuller simplifies the constant which Villarino showed was implicitly given by DeTemple and Wang.
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EXAMPLE
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3.73929751945.
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CROSSREFS
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Cf. A001008, A001620, A131915, A131916, A131918.
Sequence in context: A053010 A118452 A133056 this_sequence A019785 A074176 A005596
Adjacent sequences: A131914 A131915 A131916 this_sequence A131918 A131919 A131920
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 27 2007
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