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Search: id:A123746
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| A123746 |
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Numerators of partial sums of a series for 1/sqrt(2). |
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+0
3
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| 1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 53747636888759
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Denominators are given by A046161(n),n>=0.
The alternating sum over central binomial coefficients scaled by powers of 4, r(n):=sum(((-1)^k)*binomial(2*k,k)/4^k,k=0..n) has the limit s:=lim(r(n),n->infinity) = 1/sqrt(2). From the expansion of 1/sqrt(1-x) for |x|<1 which extends to x=-1 due to Abel's limit theorem and the convergence of the series s. See the W. Lang link.
(2^n)*n!*r(n) = A003148(n). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 06 2008]
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LINKS
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W. Lang: Rationals and more.
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FORMULA
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a(n)=numerator(r(n)) with the rationals r(n):=sum(((-1)^k)*binomial(2*k,k)/4^k,k=0..n),n>=0.
r(n)=sum(((-1)^k)*(2*k-1)!!/(2*k)!!,k=0..n),n>=0, with the double factorials A001147 and A000165.
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EXAMPLE
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a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3).
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CROSSREFS
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Cf. A120088/(2*A120777) partial sums for a series of sqrt(2).
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)
Equals A003148 divided by A049606.
(End)
Sequence in context: A087336 A137058 A116237 this_sequence A152551 A012252 A027723
Adjacent sequences: A123743 A123744 A123745 this_sequence A123747 A123748 A123749
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 10 2006
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