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A127526
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Sequence related to fifth roots of certain Fibonacci fractions.
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1
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15, 30, 91, 229, 612, 1593, 4183, 10942, 28659, 75021, 196420, 514225, 1346271, 3524574, 9227467, 24157813, 63245988, 165580137, 433494439, 1134903166, 2971215075, 7778742045, 20365011076, 53316291169, 139583862447, 365435296158, 956722026043, 2504730781957
(list; graph; refs; listen; history; internal format)
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OFFSET
| 7,1
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COMMENTS
| French examines the continued fraction expansions of the k-th roots of the fractions (Fn+k/Fn) stating [p.210]: "...something remarkable happens when k = 5. The first few terms of the continued fraction expansions for k=5 and n=1 through n=6 are listed below: [1, 1, 1, 15, 2, 2,...] [1, 1, 2, 30, 2, 3,...] [1, 1, 1, 1, 1, 91, 2, 48,...] [1, 1, 1, 1, 2, 229, 2, 12,...] [1, 1, 1, 1, 1, 1, 1, 612, 1, 1,...] [1, 1, 1, 1, 1, 1, 2, 1593, 2, 18,...] "...Fibonacci enthusiasts will have observed immediately that the sequence of large numbers one sees above, {15, 30, 91, 229, 612, 1593,...} is related to the Fibonacci sequence itself. Indeed, 15 = F7 + 2, 30 = F9 - 4, 91 = F11 + 2,...".
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REFERENCES
| Christopher P. French, "Fifth Roots of Fibonacci Fractions", The Fibonacci Quarterly, Vol. 44, No. 3; August, 2006; p. 210.
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LINKS
| Harvey P. Dale, Table of n, a(n) for n = 7..1006
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FORMULA
| F7 + 2, F9 -4, F11 + 2, F13 - 4...(F(4k - 1) + 2), (F(4k + 1) - 4)...
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EXAMPLE
| 15 = F7 + 2, 30 = F9 - 4, 91 = F11 + 2,...
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MATHEMATICA
| Total/@Partition[Riffle[Fibonacci[Range[7, 81, 2]], {2, -4}, {2, -1, 2}], 2] (* From Harvey P. Dale, Sep 20 2011 *)
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CROSSREFS
| Sequence in context: A190715 A115811 A110286 * A202522 A054305 A041442
Adjacent sequences: A127523 A127524 A127525 * A127527 A127528 A127529
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 17 2007
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EXTENSIONS
| More terms from Harvey P. Dale, Sep 20 2011
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