A214990 - OEIS

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

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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 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.

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)