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A146307 a(n) = denominator of (n-6)/(2n) = denominator of (n+6)/(2n).

%I A146307 #26 Apr 04 2024 04:35:04
%S A146307 2,1,2,4,10,1,14,8,6,5,22,4,26,7,10,16,34,3,38,20,14,11,46,8,50,13,18,
%T A146307 28,58,5,62,32,22,17,70,12,74,19,26,40,82,7,86,44,30,23,94,16,98,25,
%U A146307 34,52,106,9,110,56,38,29,118,20,122,31,42,64,130,11,134,68,46,35,142,24
%N A146307 a(n) = denominator of (n-6)/(2n) = denominator of (n+6)/(2n).
%C A146307 For numerators see A146306.
%C A146307 General formula:
%C A146307 2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =
%C A146307 Hypergeometric2F1(A146306(n)/a(n), A146306(n+12)/a(n), 1/2, 3/4).
%C A146307 2*cos(2*Pi/n) is a root of a polynomial of degree EulerPhi(n)/2 = A000010(n)/2 = A023022(n).
%C A146307 Records in this sequence are even and are congruent to 2 or 10 mod 12 (see A091999).
%C A146307 Indices where odd numbers occur in this sequence are 4n-2 (see A016825).
%C A146307 Indices where prime numbers occur in this sequence see A146309.
%C A146307 From _Robert Israel_, Apr 21 2021: (Start)
%C A146307 a(n) = 2*n if n == 1, 5, 7 or 11 (mod 12).
%C A146307 a(n) = n if n == 4 or 8 (mod 12).
%C A146307 a(n) = 2*n/3 if n == 3 or 9 (mod 12).
%C A146307 a(n) = n/2 if n == 2 or 10 (mod 12).
%C A146307 a(n) = n/3 if n == 0 (mod 12).
%C A146307 a(n) = n/6 if n == 6 (mod 12). (End)
%C A146307 Sum_{k=1..n} a(k) ~ (77/144) * n^2. - _Amiram Eldar_, Apr 04 2024
%H A146307 Robert Israel, <a href="/A146307/b146307.txt">Table of n, a(n) for n = 1..10000</a>
%H A146307 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).
%p A146307 f:= n -> denom((n-6)/(2*n)):
%p A146307 map(f, [$1..100]); # _Robert Israel_, Apr 20 2021
%t A146307 Table[Denominator[(n - 6)/(2 n)], {n, 1, 100}]
%t A146307 LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1},{2,1,2,4,10,1,14,8,6,5,22,4,26,7,10,16,34,3,38,20,14,11,46,8},80] (* _Harvey P. Dale_, May 15 2022 *)
%Y A146307 Cf. A007310, A051724, A091999, A146306 (numerators), A146308.
%K A146307 nonn,easy,frac,look
%O A146307 1,1
%A A146307 _Artur Jasinski_, Oct 29 2008