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

%I #23 Mar 30 2019 13:45:15

%S 1,1,2,2,4,6,10,16,26,41,67,108,175,283,458,740,1198,1938,3136,5074,

%T 8210,13283,21493,34776,56269,91045,147314,238358,385672,624030,

%U 1009702,1633732,2643434,4277165,6920599,11197764,18118363,29316127,47434490,76750616

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

%H Clark Kimberling, <a href="/A293633/b293633.txt">Table of n, a(n) for n = 1..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*(1 - x^3 - x^6)/((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 + 3*Fibonacci(n)/4).

%F a(n) = A293631(n) if (fractional part of 3*Fibonacci(n)/4) < 1/2, else a(n) = A293632(n).

%t z = 120; r = 3/4; f[n_] := Fibonacci[n];

%t Table[Floor[r*f[n]], {n, 1, z}]; (* A293631 *)

%t Table[Ceiling[r*f[n]], {n, 1, z}]; (* A293632 *)

%t Table[Round[r*f[n]], {n, 1, z}]; (* A293633 *)

%t LinearRecurrence[{1,1,0,0,0,1,-1,-1},{1,1,2,2,4,6,10,16},40] (* _Harvey P. Dale_, Mar 30 2019 *)

%o (PARI) a(n) = round(3*fibonacci(n)/4); \\ _Andrew Howroyd_, Feb 12 2018

%Y Cf. A000045, A293631, A293632.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Oct 14 2017

%E Offset changed by _Clark Kimberling_, Feb 12 2018