A214990 Second nearest integer to n*r, where r = (1+ sqrt(5))/2, the golden ratio.

1, 4, 4, 7, 9, 9, 12, 12, 14, 17, 17, 20, 22, 22, 25, 25, 27, 30, 30, 33, 33, 35, 38, 38, 41, 43, 43, 46, 46, 48, 51, 51, 54, 56, 56, 59, 59, 62, 64, 64, 67, 67, 69, 72, 72, 75, 77, 77, 80, 80, 82, 85, 85, 88, 88, 90, 93, 93, 96, 98, 98, 101, 101, 103, 106, 106

Offset

1,2

Comments

Let {x} denote the fractional part of x. The second nearest integer to x, denoted by s(x), is defined to be ceiling(x) if {x} < 1/2 and floor(x) if {x} >= 1/2. If x is not an integer, there are exactly two integers k such that |k-x|<1; one is round(x) = floor(x+1/2), and the other is s(x).

Let J(n) be the n-th number k for which s((k+1)*r) > s(k*r). The golden ratio appears to be the only number x for which J(n) = floor(nx) for all n>=0. In this case, J = A000201.

Let f(n) = 0 if s(n) = s(n+1) and f(n) = 1 otherwise; then f is the infinite Fibonacci word A005713 = 1-A005614.

In A214990, replace each repeated term by 1 and all others by 0; the result is A005713 (prefixed by 0).

The distinct terms of A214990 are given by A005713.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

n . . n*r . . nearest integer . second nearest
1 . . 1.618... .  2 . . . . . . . 1 = a(1)
2 . . 3.236... .  3 . . . . . . . 4 = a(2)
3 . . 4.854... .  5 . . . . . . . 4 = a(3)
4 . . 6.472... .  6 . . . . . . . 7 = a(4)
5 . . 8.090... .  8 . . . . . . . 9 = a(5)

Mathematica

r = GoldenRatio; f[x_] := If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]
Table[f[r*n], {n, 1, 100}] (* A214990 *)

Crossrefs

Cf. A214991, A000201, A005713, A005614, A001950, A005713, A007067 (complement).

Sequence in context: A107432 A198991 A371556 * A185670 A011981 A238131

Adjacent sequences: A214987 A214988 A214989 * A214991 A214992 A214993

Keyword

nonn

Author

Clark Kimberling, Oct 31 2012

Status

approved