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A309434 a(n) = floor(n*Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1)).

%I A309434 #100 Mar 08 2023 08:21:01
%S A309434 6,13,20,26,33,40,46,53,60,66,73,80,87,93,100,107,113,120,127,133,140,
%T A309434 147,154,160,167,174,180,187,194,200,207,214,220,227,234,241,247,254,
%U A309434 261,267,274,281,287,294,301,308,314,321,328,334,341
%N A309434 a(n) = floor(n*Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1)).
%C A309434 This is the Beatty sequence for Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1).
%C A309434 This is the complement of A335137.
%C A309434 Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1) = (5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)) = 6.695717525925148250774877410... = 2 + phi + tan(2*Pi/5) = A296184 + A019970.
%C A309434 For n < 10, a(n) = A109235(n).
%C A309434 Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1) = (3 + sqrt(5))/2 = 1 + phi = phi^2 = A104457.
%C A309434 Floor(n*Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1)) is A001950 (floor(n*phi^2)).
%H A309434 Karl V. Keller, Jr., <a href="/A309434/b309434.txt">Table of n, a(n) for n = 1..10000</a>
%H A309434 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence</a>.
%H A309434 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e A309434 For n = 3, floor(3*6.69571) = 20.
%t A309434 a[n_] := Floor[n * Im[2 * Exp[I * Pi/5]]/(Im[2 * Exp[I * Pi/5]] - 1)]; Array[a, 100] (* _Amiram Eldar_, Jul 06 2020 *)
%o A309434 (Python)
%o A309434 from sympy import floor, im, exp, I, pi
%o A309434 for n in range(1, 101): print(floor(n*im(2*exp(I*pi/5))/(im(2*exp(I*pi/5)) - 1)), end=', ')
%o A309434 (Python)
%o A309434 from sympy import floor, sqrt
%o A309434 for n in range(1, 101): print(floor(n*((5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)))), end=', ')
%Y A309434 Cf. A001950, A019970, A104457, A109235, A296184, A335137.
%K A309434 nonn
%O A309434 1,1
%A A309434 _Karl V. Keller, Jr._, Jun 06 2020