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 (-1 + difference sequence of A214991) 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
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}] (* A214991 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 31 2012
STATUS
approved