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Numerator of (2*n-1)/3.
9

%I #35 Apr 04 2024 04:33:12

%S 1,1,5,7,3,11,13,5,17,19,7,23,25,9,29,31,11,35,37,13,41,43,15,47,49,

%T 17,53,55,19,59,61,21,65,67,23,71,73,25,77,79,27,83,85,29,89,91,31,95,

%U 97,33,101,103,35,107,109,37,113,115,39,119,121,41,125,127,43,131,133,45

%N Numerator of (2*n-1)/3.

%C From _Jaroslav Krizek_, May 28 2010: (Start)

%C a(n+1) = numerators of antiharmonic mean of the first n positive integers for n >= 1.

%C See A169609(n-1) - denominators of antiharmonic mean of the first n positive integers for n >= 1. (End)

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).

%F From _R. J. Mathar_, Nov 21 2008: (Start)

%F a(n) = 2*a(n-3) - a(n-6).

%F G.f.: x(1+x)(1+5x^2+x^4)/((1-x)^2*(1+x+x^2)^2). (End)

%F Sum_{k=1..n} a(k) ~ (7/9) * n^2. - _Amiram Eldar_, Apr 04 2024

%e Fractions begin with 1/6, 1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 5/2, 17/6, 19/6, 7/2, 23/6, ...

%t Table[Numerator[(2 n - 1)/6], {n, 1, 100}]

%t LinearRecurrence[{0,0,2,0,0,-1},{1,1,5,7,3,11},100] (* _Harvey P. Dale_, Feb 24 2015 *)

%o (PARI) a(n) = numerator((2*n-1)/3); \\ _Altug Alkan_, Apr 13 2018

%Y Cf. A146306, A146307, A146308, A146309, A146310, A146311, A146312, A146313

%Y Trisections: A016921, A005408, A016969. [_R. J. Mathar_, Nov 21 2008]

%K nonn,easy,frac

%O 1,3

%A _Artur Jasinski_, Oct 31 2008

%E Name edited by _Altug Alkan_, Apr 13 2018