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A214315 Floor of the real part of the zeros of the complex Fibonacci function on the right half plane. 3
0, 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 54, 56, 58, 60, 62, 63, 65, 67, 69, 71, 73, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100, 102, 104, 106, 107, 109, 111, 113, 115, 117, 118 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For the complex Fibonacci function and its complex zeros see the Koshy reference, p.p. 523-4. See also the formula for F(z) given in the formula section of A052952. The real parts of the zeros of F are x_0(k) = alpha*k, with alpha:=2*(Pi^2)/(Pi^2 + (2*log(phi))^2), where phi:=(1+sqrt(5))/2, and integer k. The corresponding imaginary parts are y_0(k) = - 4*Pi*log(phi)*k/(Pi^2 + (2*log(phi))^2). alpha is approximately 1.828404783. The zeros lie in the lower right and the upper left half planes, and there is a zero at the origin.

a(n) = floor(alpha*n), n>=0, is a Beatty sequence with the complementary sequence  b(n) = floor(beta*n), with beta:= alpha/(alpha-1), approximately 2.207139336.

For the floor of the negative imaginary part see A214656.

REFERENCES

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

a(n) = floor(alpha*n), n>=0, with alpha = x_0(1) given in the comment section.

EXAMPLE

The complementary Beatty sequences start with:

n:    1   2 3  4  5  6   7   8   9  10  11  12  13  14  15  16

a(n): 0   1 3  5  7  9  10  12  14  16  18  20  21  23  25  27

b(n): (0) 2 4  6  8 11  13  15  17  19  22  24  26  28  30  33

MATHEMATICA

a[n_] := Floor[ 2*n*Pi^2 / (Pi^2 + 4*Log[GoldenRatio]^2)]; Table[a[n], {n, 0, 65}] (* Jean-Fran├žois Alcover, Jul 03 2013 *)

CROSSREFS

Cf. A214656, A052952 (F formula).

Sequence in context: A108598 A184808 A329837 * A249098 A287774 A308412

Adjacent sequences:  A214312 A214313 A214314 * A214316 A214317 A214318

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Jul 24 2012

STATUS

approved

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Last modified December 5 06:51 EST 2020. Contains 338944 sequences. (Running on oeis4.)