A146308 a(n) is the smallest k such that the numerator of (k-6)/(2k) equals n.

6, 7, 14, 15, 22, 11, 78, 13, 38, 33, 46, 17, 150, 19, 62, 51, 70, 23, 222, 25, 86, 69, 94, 29, 294, 31, 110, 87, 118, 35, 366, 37, 134, 105, 142, 41, 438, 43, 158, 123, 166, 47, 510, 49, 182, 141, 190, 53, 582, 55, 206, 159, 214, 59, 654, 61, 230, 177, 238, 65, 726

Offset: 0

Author

Artur Jasinski, Oct 29 2008

Keywords

• nonn
• look

Comments

a(n) = index of first occurrence n in A146306.

General formula:

2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) = Hypergeometric2F1(A146306(n)/A146307(n), A146306(n+12)/A146307(n), 1/2, 3/4).

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Crossrefs

Cf. A007310, A051724, A146306, A146307, A146309.

Programs

Maple

f:= proc(n) if n mod 6 = 0 then 12*n+6 elif n::even then 4*n+6 elif n mod 3 = 0 then 3*n+6 else n+6 fi end proc:
map(f, [$0..100]); # Robert Israel, Aug 05 2019

Mathematica

aa = {}; Do[k = 1; While[Numerator[(k - 6)/(2 k)] != n, k++ ]; AppendTo[aa, k], {n, 0, 100}]; aa

Formula

From Robert Israel, Aug 05 2019: (Start)

If 6 | n then a(n) = 12*n+6

else if 3 | n then a(n) = 3*n+6

else if 2 | n then a(n) = 2*n+6

else a(n) = n+6.

a(n) = 2*a(n-6) - a(n-12).

G.f.: (6 + 7*x + 14*x^2 + 15*x^3 + 22*x^4 + 11*x^5 + 66*x^6 - x^7 + 10*x^8 + 3*x^9 + 2*x^10 - 5*x^11)/(1 - 2*x^6 + x^12). (End)