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A296239 a(n) = distance from n to nearest Fibonacci number.

%I A296239 #27 Jul 02 2022 13:59:11
%S A296239 0,0,0,0,1,0,1,1,0,1,2,2,1,0,1,2,3,4,3,2,1,0,1,2,3,4,5,6,6,5,4,3,2,1,
%T A296239 0,1,2,3,4,5,6,7,8,9,10,10,9,8,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,8,9,10,
%U A296239 11,12,13,14,15,16,17,16,15,14,13,12,11,10,9
%N A296239 a(n) = distance from n to nearest Fibonacci number.
%C A296239 The Fibonacci numbers correspond to sequence A000045.
%C A296239 This sequence is analogous to:
%C A296239 - A051699 (distance to nearest prime),
%C A296239 - A053188 (distance to nearest square),
%C A296239 - A053646 (distance to nearest power of 2),
%C A296239 - A053615 (distance to nearest oblong number),
%C A296239 - A053616 (distance to nearest triangular number),
%C A296239 - A061670 (distance to nearest power),
%C A296239 - A074989 (distance to nearest cube),
%C A296239 - A081134 (distance to nearest power of 3),
%C A296239 The local maxima of the sequence correspond to positive terms of A004695.
%C A296239 a(n) = 0 iff n = A000045(k) for some k >= 0.
%C A296239 a(n) = 1 iff n = A061489(k) for some k > 4.
%C A296239 For any n >= 0, abs(a(n+1) - a(n)) <= 1.
%C A296239 For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
%C A296239 a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.
%H A296239 Harvey P. Dale, <a href="/A296239/b296239.txt">Table of n, a(n) for n = 0..1000</a>
%H A296239 <a href="/index/Di#distance_to_the_nearest">Index entries for sequences related to distance to nearest element of some set</a>
%F A296239 a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - _Jon E. Schoenfield_, Dec 14 2017
%e A296239 For n = 42:
%e A296239 - A000045(9) = 34 <= 42 <= 55 = A000045(10),
%e A296239 - a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.
%t A296239 fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* _Robert G. Wilson v_, Dec 11 2017 *)
%t A296239 With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* _Harvey P. Dale_, Jul 02 2022 *)
%o A296239 (PARI) a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))
%Y A296239 Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.
%K A296239 nonn,easy
%O A296239 0,11
%A A296239 _Rémy Sigrist_, Dec 09 2017