A214991 - OEIS

%I #10 Nov 13 2012 12:57:44

%S 2,6,7,11,14,15,19,20,23,27,28,32,35,36,40,41,44,48,49,53,54,57,61,62,

%T 66,69,70,74,75,78,82,83,87,90,91,95,96,100,103,104,108,109,112,116,

%U 117,121,124,125,129,130,133,137,138,142,143,146,150,151,155

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

%C 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.

%C Let r = golden ratio.  Then (-1 + difference sequence of A214991) consists solely of 0's, 2's, and 3's.

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

%C Positions of 2:  ([n*r^3})  A004976

%H Clark Kimberling, <a href="/A214991/b214991.txt">Table of n, a(n) for n = 1..10000</a>

%F A214991 = A000027 + A214990.

%e Let r = (3+sqrt(5))/2 = 1 + golden ratio,

%e n . . n*r . . nearest integer . second nearest

%e 1 . . 2.618... .  3 . . . . . . . 2 = a(1)

%e 2 . . 5.236... .  5 . . . . . . . 6 = a(2)

%e 3 . . 7.854... .  8 . . . . . . . 7 = a(3)

%e 4 . . 10.472.. .  10. . . . . . . 11 = a(4)

%e 5 . . 13.090.. .  13. . . . . . . 14 = a(5)

%t r = GoldenRatio^2; f[x_] := If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]

%t Table[f[r*n], {n, 1, 100}] (* A214991 *)

%Y Cf. A214990.

%K nonn

%O 1,1

%A _Clark Kimberling_, Oct 31 2012