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

1,1

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

Links

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

Formula

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

Example

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)

Mathematica

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

Crossrefs

Cf. A001950, A004976, A214990.

Sequence in context: A376131 A039568 A032926 * A286995 A088227 A231500

Adjacent sequences: A214988 A214989 A214990 * A214992 A214993 A214994

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2012

Status

approved