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A127526
Sequence related to fifth roots of certain Fibonacci fractions.
1
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
OFFSET
1,1
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, ...".
LINKS
Michel Marcus, Table of n, a(n) for n = 1..1000, after Harvey P. Dale
Jose M. Bonnin-Cadogan, Christopher P. French, and Buchan Xue, Continued Fractions of Roots of Fibonacci-like Fractions, Fibonacci Quart. 46/47 (2008/2009), no. 4, 298-311.
Christopher P. French, Fifth Roots of Fibonacci Fractions, The Fibonacci Quarterly, Vol. 44, No. 3; August, 2006; p. 210.
FORMULA
F7 + 2, F9 - 4, F11 + 2, F13 - 4, ..., (F(4k - 1) + 2), (F(4k + 1) - 4), ...
G.f.: x*(15-15*x+x^2+x^3)/((1-x)*(1+x)*(1-3*x+x^2)). - Colin Barker, Mar 09 2012
a(n) = 3*a(n-1) - 3*a(n-3) + *a(n-4) for n > 4. - Jinyuan Wang, Mar 10 2020
EXAMPLE
15 = F7 + 2, 30 = F9 - 4, 91 = F11 + 2, ...
MATHEMATICA
Total/@Partition[Riffle[Fibonacci[Range[7, 81, 2]], {2, -4}, {2, -1, 2}], 2] (* Harvey P. Dale, Sep 20 2011 *)
CoefficientList[Series[(15 - 15 x + x^2 + x^3) / ((1 - x) (1 + x) (1 - 3 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 05 2016 *)
PROG
(PARI) a(n) = if (n%2, fibonacci(2*n+5)+2, fibonacci(2*n+5)-4); \\ Michel Marcus, Mar 04 2016
(PARI) Vec(x*(15-15*x+x^2+x^3)/((1-x)*(1+x)*(1-3*x+x^2)) + O(x^100)) \\ Altug Alkan, Mar 04 2016
CROSSREFS
Sequence in context: A115811 A110286 A254859 * A249764 A202522 A054305
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jan 17 2007
EXTENSIONS
More terms from Harvey P. Dale, Sep 20 2011
Offset corrected to 1, g.f. and PARI adapted by Michel Marcus, Mar 05 2016
STATUS
approved