A127526 Sequence related to fifth roots of certain Fibonacci fractions.

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.

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

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

Adjacent sequences: A127523 A127524 A127525 * A127527 A127528 A127529

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