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A296239 a(n) = distance from n to nearest Fibonacci number. 1
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, 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, 11, 12, 13, 14, 15, 16, 17, 16, 15, 14, 13, 12, 11, 10, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

The Fibonacci numbers correspond to sequence A000045.

This sequence is analogous to:

- A051699 (distance to nearest prime),

- A053188 (distance to nearest square),

- A053646 (distance to nearest power of 2),

- A053615 (distance to nearest oblong number),

- A053616 (distance to nearest triangular number),

- A061670 (distance to nearest power),

- A074989 (distance to nearest cube),

- A081134 (distance to nearest power of 3),

The local maxima of the sequence correspond to positive terms of A004695.

a(n) = 0 iff n = A000045(k) for some k >= 0.

a(n) = 1 iff n = A061489(k) for some k > 4.

For any n >= 0, abs(a(n+1) - a(n)) <= 1.

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).

a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

LINKS

Table of n, a(n) for n=0..80.

Index entries for sequences related to distance to nearest element of some set

FORMULA

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

EXAMPLE

For n = 42:

- A000045(9) = 34 <= 42 <= 55 = A000045(10),

- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

MATHEMATICA

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 *)

PROG

(PARI) a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

CROSSREFS

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

Sequence in context: A109389 A098884 A297964 * A039800 A251418 A172101

Adjacent sequences:  A296236 A296237 A296238 * A296240 A296241 A296242

KEYWORD

nonn,easy

AUTHOR

Rémy Sigrist, Dec 09 2017

STATUS

approved

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Last modified October 20 15:22 EDT 2018. Contains 316388 sequences. (Running on oeis4.)