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

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, 28, 58, 5, 62, 32, 22, 17, 70, 12, 74, 19, 26, 40, 82, 7, 86, 44, 30, 23, 94, 16, 98, 25, 34, 52, 106, 9, 110, 56, 38, 29, 118, 20, 122, 31, 42, 64, 130, 11, 134, 68, 46, 35, 142, 24

Offset: 1

Author

Artur Jasinski, Oct 29 2008

Keywords

• nonn
• easy
• frac
• look

Comments

For numerators see A146306.

General formula:

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

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

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

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

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

Indices where prime numbers occur in this sequence see A146309.

From Robert Israel, Apr 21 2021: (Start)

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

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

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

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

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

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

Sum_{k=1..n} a(k) ~ (77/144) * n^2. - Amiram Eldar, Apr 04 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

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

Programs

Maple

f:= n -> denom((n-6)/(2*n)):
map(f, [$1..100]); # Robert Israel, Apr 20 2021

Mathematica

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