login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A015459 q-Fibonacci numbers for q=2. 10
0, 1, 1, 3, 7, 31, 143, 1135, 10287, 155567, 2789039, 82439343, 2938415279, 171774189743, 12207523172527, 1419381685547183, 201427441344229551, 46711726513354322095, 13247460522448782176431, 6135846878080826487812271 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From Gary W. Adamson, Apr 17 2009: (Start)

Preface the series with another "1": (1, 1, 1, 3, 7, ..., a(n)).

Then a(n+2) = (1, 1, 1, 3, 7, ..., a(n)) dot (1, 2, 4, 8, ...).

Example: (143) = (1, 1, 1, 3, 7) dot (1, 2, 4, 8, 16) = (1 + 2 + 4 + 24 + 112).

Analogous procedures apply to other q-Fibonacci sequences for q(n). (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100.

L. Carlitz, Fibonacci notes 3: q-Fibonacci Numbers, Fibonacci Quarterly 12 (1974), pp. 317-322.

Eric Weisstein's World of Mathematics, q-Analog.

FORMULA

a(n) = a(n-1) + 2^(n-2) a(n-2).

Associated constant: C_2 = lim n ->infty a(2*n)*a(2*n-2)/a(2*n-1)^2 = 1.225306147422043724739386133....(C_1=1). - Benoit Cloitre, Aug 30 2003 [Formula corrected by Vaclav Kotesovec, Dec 05 2017]

a(n)*a(n+3)-a(n)*a(n+2)-2*a(n+1)*a(n+2)+2*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017

From Vaclav Kotesovec, Dec 05 2017: (Start)

a(n) ~ c * 2^(n*(n-2)/4), where

c = 2.815179026313038425026160599838001991828247939843695... if n is even and

c = 3.024413799639405763259604599843170276573526808693115... if n is odd. (End)

MATHEMATICA

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*2^(n-2)}, a, {n, 20}] (* Vincenzo Librandi, Nov 08 2012 *)

PROG

(MAGMA) [0] cat [n le 2 select 1 else Self(n-1) + Self(n-2)*(2^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

(Sage) from ore_algebra import *; R.<x> = QQ['x']; A.<Qx> = OreAlgebra(R, 'Qx', q=2); print (Qx^2 - Qx - x).to_list([0, 1], 10)  # Ralf Stephan, Apr 24 2014

(Python)

def a():

    a, b, p = 0, 1, 1

    while True:

        yield a

        p, a, b = p + p, b, b + p * a

A015463 = a()

print([next(A015463) for _ in range(20)]) # Peter Luschny, Dec 05 2017

CROSSREFS

Cf. A015460.

Sequence in context: A121620 A042271 A000644 * A115083 A141385 A059296

Adjacent sequences:  A015456 A015457 A015458 * A015460 A015461 A015462

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified January 20 01:27 EST 2018. Contains 297938 sequences.