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%I A214991 #20 Oct 28 2024 09:34:38
%S A214991 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 A214991 66,69,70,74,75,78,82,83,87,90,91,95,96,100,103,104,108,109,112,116,
%U A214991 117,121,124,125,129,130,133,137,138,142,143,146,150,151,155
%N A214991 Second nearest integer to n*(1+golden ratio).
%C A214991 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 A214991 Let r = golden ratio.  Then (a(n+1) - a(n) - 1) consists solely of 0's, 2's, and 3's.
%C A214991 Positions of 0:  ([n*r^2])  A001950
%C A214991 Positions of 2:  ([n*r^3])  A004976
%H A214991 Clark Kimberling, <a href="/A214991/b214991.txt">Table of n, a(n) for n = 1..10000</a>
%F A214991 a(n) = n + A214990(n).
%e A214991 Let r = (3+sqrt(5))/2 = 1 + golden ratio,
%e A214991 n . . n*r . . nearest integer . second nearest
%e A214991 1 . . 2.618... .  3 . . . . . . . 2 = a(1)
%e A214991 2 . . 5.236... .  5 . . . . . . . 6 = a(2)
%e A214991 3 . . 7.854... .  8 . . . . . . . 7 = a(3)
%e A214991 4 . . 10.472.. .  10. . . . . . . 11 = a(4)
%e A214991 5 . . 13.090.. .  13. . . . . . . 14 = a(5)
%t A214991 r = GoldenRatio^2; f[x_] := If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]
%t A214991 Table[f[r*n], {n, 1, 100}]
%Y A214991 Cf. A001950, A004976, A214990.
%K A214991 nonn,easy,changed
%O A214991 1,1
%A A214991 _Clark Kimberling_, Oct 31 2012

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