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Search: id:A124397
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| A124397 |
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Numerators of partial sums of a series for sqrt(5)/3. |
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+0
2
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| 1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279
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OFFSET
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0,2
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COMMENT
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Denominators are given by A124398.
The alternating sums over central binomial coefficients scaled by powers of 5, r(n):=sum(((-1)^k)*binomial(2*k,k)/5^k,k=0..n) have the limit s:=lim(r(n),n->infinity) = sqrt(5)/3. 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(((-1)^k)*binomial(2*k,k)/5^k,k=0..n) in lowest terms.
r(n)=sum(((-1)^k)*((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)=17 because r(3)= 1-2/5+6/25-4/25 = 17/25 = a(3)/A124398(3).
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CROSSREFS
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Cf. A123747/A123748 partial sums for a series for sqrt(5).
Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).
Sequence in context: A024011 A052445 A089999 this_sequence A043081 A091675 A067233
Adjacent sequences: A124394 A124395 A124396 this_sequence A124398 A124399 A124400
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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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