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

Author

Clark Kimberling, Oct 31 2012

Keywords

• nonn

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 a(n) = a(n+1) and f(n) = 1 otherwise; then f is the infinite Fibonacci word A005614 = 1-A003849.

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

The distinct terms of this sequence are given by A007064.

Examples

			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)
		

Links

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

Crossrefs

Cf. A214991, A000201, A003849, A005713, A005614, A007064, A001950, A007067 (complement).

Programs

Mathematica

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