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Search: id:A123749
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| A123749 |
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Numerators of partial sums of a series for 3/sqrt(5)=(3/5)*sqrt(5). |
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3
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| 1, 11, 35, 965, 8755, 8783, 237449, 2138185, 6415985, 519743405, 4677875401, 14033861347, 378916960525, 3410263045325, 3410267502725, 30692424759805, 276231889624955, 828695755304725, 67124359204727825, 604119244624305025
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Denominators are given by A124396.
The sums over central binomial coefficients scaled by powers of 9, r(n):=sum(binomial(2*k,k)/9^k,k=0..n) have the limit s:=lim(r(n),n->infinity) = 3/sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/9.
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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(binomial(2*k,k)/9^k,k=0..n) in lowest terms.
r(n)=sum(((2*k-1)!!/((2*k)!!)*(4/9)^k,k=0..n),n>=0, with the double factorials A001147 and A000165.
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EXAMPLE
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a(3)=965 because r(3)=1+2/9+2/27+20/729 = 965/729 = a(3)/A124396(3).
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CROSSREFS
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Cf. A123747/A123748 partial sums for a series for sqrt(5).
Sequence in context: A103115 A003777 A098116 this_sequence A159493 A012644 A138893
Adjacent sequences: A123746 A123747 A123748 this_sequence A123750 A123751 A123752
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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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