A264906 a(n) is the denominator of the 2nd term of the power series which is the loop length in a regular n-gon. (See comment.)

25, 36, 49, 64, 81, 100, 121, 72, 169, 196, 225, 256, 289, 324, 361, 100, 441, 484, 529, 576, 625, 676, 729, 392, 841, 900, 961, 1024, 1089, 1156, 1225, 324, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 968, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 676, 2809

Offset

5,1

Comments

Inspired by A262343. Given a regular n-gon whose sides are of unit length, draw around each vertex V a circular arc connecting vertex V's two next-to-nearest neighbors. Connect the n arcs thus drawn into a single closed curve if n is odd, or a pair of identical (but rotated by 1/n of a turn) closed curves if n mod 4 = 2, or four identical (but rotated by 1/n of a turn) closed curves if n mod 4 = 0. (See illustration in Links.)

The values of the loop length L(n) appear to form a power series. Conjectures: the coefficient of the first term is 2*A060819; the numerator and denominator of the coefficient of the 2nd term are -1*A000265 and a(n), respectively; and the numerator of the coefficient of the 3rd term is A109375.

Links

Table of n, a(n) for n=5..53.

Kival Ngaokrajang, Illustration of loop length L(n) for n = 5..12

Formula

a(n) = n^2/gcd((n-4)/gcd(n-4,4),n); for n >= 5.

Example

L(5) = 2*Pi - 1/25*Pi^3 + 1/7500*Pi^5 - 1/5625000*Pi^7 + 1/7875000000*Pi^9 - ...
L(6) = 2*Pi - 1/36*Pi^3 + 1/15552*Pi^5 - 1/16796160*Pi^7 + 1/33861058560*Pi^9 - ...
L(7) = 6*Pi - 3/49*Pi^3 + 1/9604*Pi^5 - 1/14117880*Pi^7 + 1/38739462720*Pi^9 - ...
L(8) = 2*Pi - 1/64*Pi^3 + 1/49152*Pi^5 - 1/94371840*Pi^7 + 1/338228674560*Pi^9 - ...
L(9) = 10*Pi - 5/81*Pi^3 + 5/78732*Pi^5 - 1/38263752*Pi^7 + 1/173564379072*Pi^9 - ...
L(10) = 6*Pi - 3/100*Pi^3 + 1/40000*Pi^5 - 1/120000000*Pi^7 + 1/672000000000*Pi^9 - ...
...
Let T(n) be the total of the loop lengths, i.e., T(n) = L(n) if n is odd, 2*L(n) if n mod 4 = 2, and 4*L(n) if n mod 4 = 0. Multiplying each of the above series expansions for L(n) by the appropriate multiplier (i.e., 1, 2, or 4) to get T(n) gives expansions for L(5)..L(10) that agree with the general form
T(n) = 2*(n-4) * Sum_{k>=0} (-1)^k * Pi^(2k+1) / ((2k)! * n^(2k)) for n=5..10.

Mathematica

Table[n^2/GCD[(n - 4)/GCD[n - 4, 4], n], {n, 5, 46}] (* Michael De Vlieger, Nov 28 2015 *)

Prog

(PARI) {for(n = 5, 100, k = 1; if (Mod(n, 4)==0, k = 4); if (Mod(n, 4)==2, k = 2); arc = 2*cos(x/n)*x*(1-4/n); loop = n*arc/k; print(loop))} \\ L(n)
(PARI) {for(n = 5, 100, a = n^2/gcd((n-4)/gcd(n-4, 4), n); print1(a, ", "))} \\ a(n)
(Magma) [n^2 div Gcd((n-4) div Gcd(n-4, 4), n): n in [5..60]]; // Vincenzo Librandi, Nov 29 2015

Crossrefs

Cf. A000265, A060819, A109375, A262343.

Sequence in context: A077689 A018796 A029945 * A374935 A263095 A077502

Adjacent sequences: A264903 A264904 A264905 * A264907 A264908 A264909

Keyword

nonn,frac

Author

Kival Ngaokrajang, Nov 28 2015

Extensions

More terms from Vincenzo Librandi, Nov 29 2015

Status

approved