7,1

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,...".

Christopher P. French, "Fifth Roots of Fibonacci Fractions", The Fibonacci Quarterly, Vol. 44, No. 3; August, 2006; p. 210.

F7 + 2, F9 -4, F11 + 2, F13 - 4...(F(4k - 1) + 2), (F(4k + 1) - 4)...

15 = F7 + 2, 30 = F9 - 4, 91 = F11 + 2,...

Sequence in context: A072304 A115811 A110286 this_sequence A054305 A041442 A041440

Adjacent sequences: A127523 A127524 A127525 this_sequence A127527 A127528 A127529

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 17 2007