A214991 Second nearest integer to n*(1+golden ratio).

2, 6, 7, 11, 14, 15, 19, 20, 23, 27, 28, 32, 35, 36, 40, 41, 44, 48, 49, 53, 54, 57, 61, 62, 66, 69, 70, 74, 75, 78, 82, 83, 87, 90, 91, 95, 96, 100, 103, 104, 108, 109, 112, 116, 117, 121, 124, 125, 129, 130, 133, 137, 138, 142, 143, 146, 150, 151, 155

Offset: 1

Author

Clark Kimberling, Oct 31 2012

Keywords

• nonn
• easy

Comments

Let {x} denote the fractional part of x. The second nearest integer to x is defined to be ceiling(x) if {x}<1/2 and floor(x) if {x}>=1/2.

Let r = golden ratio. Then (a(n+1) - a(n) - 1) consists solely of 0's, 2's, and 3's.

Positions of 0: ([n*r^2]) A001950

Positions of 2: ([n*r^3]) A004976

Examples

			Let r = (3+sqrt(5))/2 = 1 + golden ratio,
n . . n*r . . nearest integer . second nearest
1 . . 2.618... .  3 . . . . . . . 2 = a(1)
2 . . 5.236... .  5 . . . . . . . 6 = a(2)
3 . . 7.854... .  8 . . . . . . . 7 = a(3)
4 . . 10.472.. .  10. . . . . . . 11 = a(4)
5 . . 13.090.. .  13. . . . . . . 14 = a(5)
		

Links

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

Crossrefs

Cf. A001950, A004976, A214990.

Programs

Mathematica

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

Formula

a(n) = n + A214990(n).