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A154925
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The terms of this sequence are integer values of consecutive denominators (with signs) from the fractional expansion (using only fractions with numerators to be positive 1's) of the BBP polynomial ( 4/(8*k+1) - 2/(8*k+4) - 1/(8*k+5) - 1/(8*k+6) ) for all k (starting from 0 to infinity).
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3
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1, 1, 1, 1, -2, -5, -6, 3, 9, -5, -13, -14, 5, 30, 510, -10, -21, -22, 7, 59, 5163, 53307975, -14, -29, -30
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OFFSET
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0,5
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COMMENTS
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The Egyptian fraction expansion is applied to the first fraction (that is, 4/(8*k+1) ) of the BBP polynomial ( 4/(8*k+1) - 2/(8*k+4) - 1/(8*k+5) - 1/(8*k+6) ) for k >= 1. R. Knott's converter calculator #1 (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#calc1) is used for such conversion. Note that in the case of k=0, 4/(8*k+1) = 4 and could be trivially expressed as 1/1 + 1/1 + 1/1 + 1/1. It remains to be seen how the above described Pi presentation relates to Engel's presentation of Pi, which also consists of an infinite sum of fractions whose numerators are all 1's.
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LINKS
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EXAMPLE
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For k=1, 4/(8*k+1) = 4/9 = 1/3 + 1/9, thus the first (smallest) denominator is 3 so a(7)=3.
For k=1, 4/(8*k+1) = 4/9 = 1/3 + 1/9 and the second (next to smallest) denominator is 9 so a(8)=9.
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CROSSREFS
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KEYWORD
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sign,uned
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AUTHOR
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STATUS
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approved
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