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A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.

%I A293553 #12 Oct 22 2018 08:29:01
%S A293553 0,0,0,0,1,1,2,3,5,8,14,22,36,58,94,152,247,399,646,1045,1691,2736,
%T A293553 4428,7164,11592,18756,30348,49104,79453,128557,208010,336567,544577,
%U A293553 881144,1425722,2306866,3732588,6039454,9772042,15811496,25583539,41395035,66978574
%N A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.
%H A293553 Clark Kimberling, <a href="/A293553/b293553.txt">Table of n, a(n) for n = 0..1000</a>
%H A293553 <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 A293553 G.f.: x^4/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2)).
%F A293553 a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.
%F A293553 a(n) = floor(1/2 + Fibonacci(n)/4).
%F A293553 a(n) = A004697(n) if (fractional part of Fibonacci(n)/4) < 1/2, otherwise a(n) = A293552(n).
%F A293553 a(n) = A131132(n-4) for n > 3. - _Georg Fischer_, Oct 22 2018
%t A293553 z = 120; r = 1/4; f[n_] := Fibonacci[n];
%t A293553 Table[Floor[r*f[n]], {n, 0, z}];   (* A004697 *)
%t A293553 Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293552 *)
%t A293553 Table[Round[r*f[n]], {n, 0, z}];   (* A293553 *)
%Y A293553 Cf. A000045, A004697, A131132, A293552.
%K A293553 nonn,easy
%O A293553 0,7
%A A293553 _Clark Kimberling_, Oct 14 2017