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A214991
Second nearest integer to n*(1+golden ratio).
2
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
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
KEYWORD
nonn,easy,changed
AUTHOR
Clark Kimberling, Oct 31 2012
STATUS
approved