A146307 - OEIS

%I #26 Apr 04 2024 04:35:04

%S 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 28,58,5,62,32,22,17,70,12,74,19,26,40,82,7,86,44,30,23,94,16,98,25,

%U 34,52,106,9,110,56,38,29,118,20,122,31,42,64,130,11,134,68,46,35,142,24

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

%C For numerators see A146306.

%C General formula:

%C 2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =

%C Hypergeometric2F1(A146306(n)/a(n), A146306(n+12)/a(n), 1/2, 3/4).

%C 2*cos(2*Pi/n) is a root of a polynomial of degree EulerPhi(n)/2 = A000010(n)/2 = A023022(n).

%C Records in this sequence are even and are congruent to 2 or 10 mod 12 (see A091999).

%C Indices where odd numbers occur in this sequence are 4n-2 (see A016825).

%C Indices where prime numbers occur in this sequence see A146309.

%C From _Robert Israel_, Apr 21 2021: (Start)

%C a(n) = 2*n if n == 1, 5, 7 or 11 (mod 12).

%C a(n) = n if n == 4 or 8 (mod 12).

%C a(n) = 2*n/3 if n == 3 or 9 (mod 12).

%C a(n) = n/2 if n == 2 or 10 (mod 12).

%C a(n) = n/3 if n == 0 (mod 12).

%C a(n) = n/6 if n == 6 (mod 12). (End)

%C Sum_{k=1..n} a(k) ~ (77/144) * n^2. - _Amiram Eldar_, Apr 04 2024

%H Robert Israel, <a href="/A146307/b146307.txt">Table of n, a(n) for n = 1..10000</a>

%H <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 f:= n -> denom((n-6)/(2*n)):

%p map(f, [$1..100]); # _Robert Israel_, Apr 20 2021

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

%t 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 Cf. A007310, A051724, A091999, A146306 (numerators), A146308.

%K nonn,easy,frac,look

%O 1,1

%A _Artur Jasinski_, Oct 29 2008