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A145375
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Numerators of partial sums of the alternating series of inverse central binomial coefficients.
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4
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1, 1, 23, 31, 47, 1031, 26827, 134107, 455989, 8663665, 4331849, 187279, 4981622687, 747243353, 173360460899, 1074834852769, 233659750871, 926581770421, 198844447947463, 6856705101503, 1630524473145553, 350562761725846217, 97378544923877951, 42247307182355837
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| See A145556 for the denominators.
The limit of the rational partial sums r(n), defined below, for n->infinity is (1 + 4*ln(phi)/(2*phi-1))/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.3721635764.
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REFERENCES
| C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. Eq.13, p.39.
A. J. van der Poorten, Some wonderful formulae...Footnote to Apery's proof of the irrationality of zeta(3), S\'eminaire Delange-Pisot-Poitou. Th\'eorie des nombres, tome 20, no 2 (1978-1979), exp, no 29, p.1-7. p. 29-02 Available via http://www.numdam.org/numdam-bin/qrech
R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.
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LINKS
| W. Lang, Rationals and more.
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FORMULA
| a(n)=numerator(r(n)) with the rationals (in lowest terms) r(n):=sum(((-1)^(k+1))/binomial(2*k,k),k=1..n).
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EXAMPLE
| Rationals r(n) (in lowest terms): [1/2,1/3,23/60,31/84,47/126,1031/2772,26827/72072,...].
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CROSSREFS
| A145557/A145558.
Sequence in context: A162587 A033216 A139837 * A086547 A054291 A052230
Adjacent sequences: A145372 A145373 A145374 * A145376 A145377 A145378
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 17 2008, Nov 17 2008, Nov 25 2008
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