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A131918
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Continued fraction expansion of 1 / (1 - gamma - ln(3/2)) - 54, where gamnma is the Euler-Mascheroni constant.
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3
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3, 1, 2, 1, 5, 11, 7, 6, 1, 2, 6, 1, 10, 15, 7, 1, 11, 12, 1, 1, 4, 3, 1, 1, 9, 3, 4, 10, 4, 1, 1, 26, 1, 1, 8, 10, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 3, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 4, 2, 1, 49, 7, 1, 2, 1, 1, 2, 16, 1, 283, 1, 1, 5, 1, 1, 1, 2, 1, 30, 19, 1, 11, 2, 5, 10, 3, 1, 4, 1, 6, 2, 19, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Decimal expansion is A131917. 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
| 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
| (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
| 3.73929751945... = 3 + 1/1+ 1/2+ 1/1+ 1/5+ 1/11+ 1/7+ 1/6+ 1/1+ 1/2+ 1/6+ 1/1+ 1/10+ 1/15+ 1/7+ 1/1+ 1/11+ 1/12+ 1/1+ 1/1+ 1/4+ 1/3+ 1/1+ 1/1+ 1/9+ 1/3+ 1/4+ 1/10+ 1/4+ 1/1+ 1/1+ 1/26+ 1/1+ 1/1+ 1/8+ 1/10+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/2+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/3+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/4+ 1/2+ 1/1+ 1/49+ 1/7+ 1/1+ 1/2+ 1/1+ 1/1+ 1/2+ 1/16+ 1/1+ 1/283+ 1/1+ 1/1+ 1/5+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/30+ 1/19+ 1/1+ 1/11+ 1/2+ 1/5+ 1/10+ 1/3+ 1/1+ 1/4+ 1/1+ 1/6+ 1/2+ 1/19+ 1/1+ 1/1+ 1/3+ 1/2+ 1/1+ . . .
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CROSSREFS
| Cf. A001008, A001620, A131915, A131916, A131917.
Sequence in context: A059807 A135261 A102774 * A010123 A039620 A008296
Adjacent sequences: A131915 A131916 A131917 * A131919 A131920 A131921
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KEYWORD
| cofr,easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 27 2007
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