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A293420 a(n) is the integer k that minimizes |k/Fibonacci(n) - sqrt(2)|.

%I A293420 #12 Sep 08 2022 08:46:19
%S A293420 0,1,1,3,4,7,11,18,30,48,78,126,204,330,533,863,1396,2258,3654,5913,
%T A293420 9567,15480,25047,40527,65574,106101,171676,277777,449453,727230,
%U A293420 1176682,1903912,3080594,4984506,8065100,13049606,21114706,34164312,55279019,89443331
%N A293420 a(n) is the integer k that minimizes |k/Fibonacci(n) - sqrt(2)|.
%H A293420 Clark Kimberling, <a href="/A293420/b293420.txt">Table of n, a(n) for n = 0..1000</a>
%F A293420 a(n) = floor(1/2 + Fibonacci(n)*sqrt(2)).
%F A293420 a(n) = A293418(n) if (fractional part of Fibonacci(n)*sqrt(2)) < 1/2, otherwise a(n) = A293419(n).
%t A293420 z = 120; r = Sqrt[2]; f[n_] := Fibonacci[n];
%t A293420 Table[Floor[r*f[n]], {n, 0, z}];   (* A293418 *)
%t A293420 Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293419 *)
%t A293420 Table[Round[r*f[n]], {n, 0, z}];   (* A293420 *)
%o A293420 (PARI) for(n=0,30, print1(round(fibonacci(n)*sqrt(2)), ", ")) \\ _G. C. Greubel_, Feb 08 2018
%o A293420 (Magma) [Round(Fibonacci(n)*Sqrt(2)): n in [0..30]]; // _G. C. Greubel_, Feb 08 2018
%Y A293420 Cf. A000045, A293418, A293419.
%K A293420 nonn,easy
%O A293420 0,4
%A A293420 _Clark Kimberling_, Oct 12 2017