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A146308
a(n) is the smallest k such that the numerator of (k-6)/(2k) equals n.
5
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
0,1
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).
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)
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
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Artur Jasinski, Oct 29 2008
STATUS
approved