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a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/2|.
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%I #7 Feb 17 2018 20:04:40

%S 0,0,0,1,2,2,4,6,10,17,28,44,72,116,188,305,494,798,1292,2090,3382,

%T 5473,8856,14328,23184,37512,60696,98209,158906,257114,416020,673134,

%U 1089154,1762289,2851444,4613732,7465176,12078908,19544084,31622993,51167078

%N a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/2|.

%H Clark Kimberling, <a href="/A293505/b293505.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 1, -1, -1)

%F G.f.: -((x^3 (-1 - x + x^2))/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2))).

%F a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.

%F a(n) = floor(1/2 + Fibonacci(n)/2).

%F a(n) = A004695(n) if (fractional part of Fibonacci(n)/2) < 1/2, otherwise a(n) = A293419(n).

%t z = 120; r = 1/2; f[n_] := Fibonacci[n];

%t Table[Floor[r*f[n]], {n, 0, z}]; (* A004695 *)

%t Table[Ceiling[r*f[n]], {n, 0, z}]; (* A173173 *)

%t Table[Round[r*f[n]], {n, 0, z}]; (* A293505 *)

%Y Cf. A000045, A004695, A293505.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Oct 12 2017