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a(n) is the integer k that minimizes |k/Fibonacci(n) - 4/5|.
3

%I #7 Feb 17 2018 20:06:32

%S 0,1,1,2,2,4,6,10,17,27,44,71,115,186,302,488,790,1278,2067,3345,5412,

%T 8757,14169,22926,37094,60020,97114,157134,254249,411383,665632,

%U 1077015,1742647,2819662,4562310,7381972,11944282,19326254,31270535,50596789,81867324

%N a(n) is the integer k that minimizes |k/Fibonacci(n) - 4/5|.

%H Clark Kimberling, <a href="/A293673/b293673.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, -1, -2, 1, 2, -1, -2, 1, 1)

%F G.f.: -((x (1 - x^2 - x^3 + x^4 + x^5 - x^6 + x^8))/((-1 + x + x^2) (1 - x^2 + x^4 - x^6 + x^8))).

%F a(n) = a(n-1) + 2 a(n-2) - a(n-3) - 2 a(n-4) + a(n-5) + 2 a(n-6) - a(n-7) - 2 a(n-8) + a(n-9) + a(n-10) for n >= 11.

%F a(n) = floor(1/2 + 4*Fibonacci(n)/5).

%F a(n) = A293671(n) if (fractional part of 4*Fibonacci(n)/5) < 1/2, otherwise a(n) = A293672(n).

%t z = 120; r = 4/5; f[n_] := Fibonacci[n];

%t Table[Floor[r*f[n]], {n, 0, z}]; (* A293671 *)

%t Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293672 *)

%t Table[Round[r*f[n]], {n, 0, z}]; (* A293673 *)

%Y Cf. A000045, A293671, A293672.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Oct 16 2017