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Search: id:A123747
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| A123747 |
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Numerators of partial sums of a series for sqrt(5). |
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+0
6
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| 1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583
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OFFSET
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0,2
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COMMENT
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Denominators are given by A123748.
The sum over central binomial coefficients scaled by powers of 5, r(n):=sum(binomial(2*k,k)/5^k,k=0..n) has the limit s:=lim(r(n),n->infinity) = sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/5.
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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)/5^k,k=0..n) in lowest terms.
r(n)=sum(((2*k-1)!!/((2*k)!!)*(4/5)^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(3)= 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).
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CROSSREFS
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Cf. A001077/A001076 continued fraction convergents for sqrt(5).
Sequence in context: A121582 A062727 A165397 this_sequence A144421 A023251 A073501
Adjacent sequences: A123744 A123745 A123746 this_sequence A123748 A123749 A123750
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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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