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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, pp. 523-524. 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 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-François 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 3 12:44 EST 2021. Contains 349463 sequences. (Running on oeis4.)