A262343 Numerator of 3*(1-2/n), for n >= 3.

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, 39, 81, 14, 87, 45, 31, 48, 99, 17, 105, 54, 37, 57, 117, 20, 123, 63, 43, 66, 135, 23, 141, 72, 49, 75, 153, 26, 159, 81, 55, 84, 171, 29, 177, 90, 61, 93, 189, 32

Offset: 3

Author

Kival Ngaokrajang, Sep 18 2015

Keywords

• nonn
• easy
• frac

Comments

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:

(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.

(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).

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

Links

• Peter Kagey, Table of n, a(n) for n = 3..10000
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Crossrefs

Cf. A060789, A026741.

Programs

Magma

[Numerator(3*(1-2/n)): n in [3..80]]; // Vincenzo Librandi, Sep 19 2015

Maple

a:= proc(n): numer(3*(n-2)/n) end: seq(a(n), n=3..66); # Johannes W. Meijer, Jul 03 2016

Mathematica

Table[Numerator[3 (1 - 2/n)], {n, 3, 60}] (* Michael De Vlieger, Sep 18 2015 *)

PARI

{for(n=3, 100, a=numerator(3*(1-2/n)); print1 (a, ", "))}

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

Formula

a(n) = numerator(3*(1-2/n)), for n >= 3.

From Peter Kagey, Sep 18 2015: (Start)

For integers k:

a(6k+0) = 3 * k - 1

a(6k+1) = 18 * k - 3

a(6k+2) = 9 * k + 1

a(6k+3) = 6 * k + 1

a(6k+4) = 9 * k + 3

a(6k+5) = 18 * k + 9

(End)

From Colin Barker, Sep 20 2015: (Start)

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

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).

(End)

Extensions

More terms from Vincenzo Librandi, Sep 19 2015