%I #49 Sep 08 2022 08:46:14
%S 1,3,9,2,15,9,7,12,27,5,33,18,13,21,45,8,51,27,19,30,63,11,69,36,25,
%T 39,81,14,87,45,31,48,99,17,105,54,37,57,117,20,123,63,43,66,135,23,
%U 141,72,49,75,153,26,159,81,55,84,171,29,177,90,61,93,189,32
%N Numerator of 3*(1-2/n), for n >= 3.
%C Given a regular n-gon with side length s, draw a circular arc of radius s around each of the n-gon's vertices so as to connect that vertex's two nearest neighbors, drawing the arc on the shorter side of the circle; i.e., each arc will extend through an angle of Pi*(n-2)/n radians (see illustration). Connect the n arcs thus drawn into a single closed curve if n is odd, or into a pair of identical (but with one rotated by 2*Pi/n radians with respect to the other) overlapping closed curves if n is even. The arcs and the curve (or pair of curves) have the following properties:
%C (i) Since the length L(n) of each single arc is L(n) = s*Pi*(n-2)/n, the ratio of the length of a single arc for an n-gon to the length of a single arc for the n=3 case is L(n)/L(3) = (s*Pi*(n-2)/n)/(s*Pi*(3-2)/3) = 3(1-2/n). The numerator and denominator of 3(1-2/n) are a(n) and A060789(n) respectively.
%C (ii) Since the loop length (considering only one of the two loops when there are two overlapping loops) is L(n)*n when n is odd, or L(n)*n/2 when n is even, the ratio of the loop length for an n-gon to the loop length for the n=3 case is (L(n)*n)/(L(3)*3) = (s*Pi*(n-2))/(s*Pi) = n-2 when n is odd, or (L(n)*n/2)/(L(3)*3) = (s*Pi*(n-2)/2)/(s*Pi) = (n-2)/2 when n is even; thus, whether odd or even, that ratio is numerator(1-2/n) = A026741(n-2).
%C The moment generating function of p(x, m=1, n=2, mu=2) = 3*x*E(x, 1, 2), see A163931 and A274181, is given by M(a) = (3*a-6)/(a^2*(a-1)) + 6*log(1-a)/a^3. The series expansion of M(a) leads to the sequence given above. - _Johannes W. Meijer_, Jul 04 2016
%H Peter Kagey, <a href="/A262343/b262343.txt">Table of n, a(n) for n = 3..10000</a>
%H Kival Ngaokrajang, <a href="/A262343/a262343_2.pdf">Illustration of initial terms</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
%F a(n) = numerator(3*(1-2/n)), for n >= 3.
%F From _Peter Kagey_, Sep 18 2015: (Start)
%F For integers k:
%F a(6k+0) = 3 * k - 1
%F a(6k+1) = 18 * k - 3
%F a(6k+2) = 9 * k + 1
%F a(6k+3) = 6 * k + 1
%F a(6k+4) = 9 * k + 3
%F a(6k+5) = 18 * k + 9
%F (End)
%F From _Colin Barker_, Sep 20 2015: (Start)
%F a(n) = 2*a(n-6) - a(n-12).
%F G.f.: x^3*(3*x^10+x^9+9*x^8+6*x^7+5*x^6+9*x^5+15*x^4+2*x^3+9*x^2+3*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2).
%F (End)
%p a:= proc(n): numer(3*(n-2)/n) end: seq(a(n), n=3..66); # _Johannes W. Meijer_, Jul 03 2016
%t Table[Numerator[3 (1 - 2/n)], {n, 3, 60}] (* _Michael De Vlieger_, Sep 18 2015 *)
%o (PARI) {for(n=3, 100, a=numerator(3*(1-2/n)); print1 (a, ", "))}
%o (Magma) [Numerator(3*(1-2/n)): n in [3..80]]; // _Vincenzo Librandi_, Sep 19 2015
%o (PARI) Vec(x^3*(3*x^10+x^9+9*x^8+6*x^7+5*x^6+9*x^5+15*x^4+2*x^3+9*x^2+3*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Sep 20 2015
%Y Cf. A060789, A026741.
%K nonn,easy,frac
%O 3,2
%A _Kival Ngaokrajang_, Sep 18 2015
%E More terms from _Vincenzo Librandi_, Sep 19 2015