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Search: id:A120088
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| A120088 |
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Numerators of partial sums of a series for sqrt(2). |
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+0
3
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| 3, 11, 23, 179, 365, 1439, 2911, 46147, 93009, 369605, 743409, 5917879, 11887761, 47365319, 95064943, 3032383331, 6082445497, 24264959593, 48649328861, 388310999293, 778263028691, 3106935548009, 6225306416473
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k.
From the expansion of sqrt(1+x) = 1 + x*sum((C_k)*(-x/4)^k,k=0..infty)/2, valid for |x|<=1, one finds for x=+1: sqrt(2) = 1 + sum(((-1)^k)*C(k)/4^k,k=0..infty)/2.
The denominators are given by 2*A120777(n).
The rationals r(n):=1 + (sum(((-1)^k)*C(k)/4^k))/2,k=0..n), with the Catalan numbers C(n)=A000108(n), are A120088(n)/ A120777(n),n>=0.
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LINKS
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W. Lang: Rationals r(n).
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FORMULA
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a(n)=numerator(r(n)), with the rationals defined above.
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EXAMPLE
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Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024,
2911/2048, 46147/32768,...]
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CROSSREFS
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For similar partial sums with positive terms (not alternating) see rationals A119951/A120069.
For the partial sums (sum(((-1)^k)*C(k)/4^k)), k=0..n) see A120788(n)/A120777(n).
Sequence in context: A002515 A096297 A081857 this_sequence A081737 A005475 A112051
Adjacent sequences: A120085 A120086 A120087 this_sequence A120089 A120090 A120091
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 2006
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