A262343 -id:A262343 - cp's OEIS frontend

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

Kival Ngaokrajang, Nov 28 2015

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.

			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.

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

Cf. A000265, A060819, A109375, A262343.

[n^2 div Gcd((n-4) div Gcd(n-4,4),n): n in [5..60]]; // Vincenzo Librandi, Nov 29 2015

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

{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)

{for(n = 5, 100, a = n^2/gcd((n-4)/gcd(n-4,4), n); print1(a, ", "))} \\ a(n)

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

More terms from Vincenzo Librandi, Nov 29 2015

A266685 T(n,k) is the number of loops appearing in pattern of circular arc connecting two vertices of regular polygons. (See Comments.)

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 3, 2, 1, 6, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 18

Offset: 0

Kival Ngaokrajang, Jan 02 2016

The patterns in A262343 and A264906 can be considered as case of skip 0 and 1 vertex of circle construction on regular polygons. k is the cyclic number of loops of the case skip n-vertices. See illustration for more details.

T(n,k) is conjectured to be even rows of A109004 (excluding the first column).

			Irregular triangle begins:
n\k   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
0     1  2
1     1  2  1  4
2     1  2  3  2  1  6
3     1  2  1  4  1  2  1  8
4     1  2  1  2  5  2  1  2  1 10
5     1  2  3  4  1  6  1  4  3  2  1 12
6     1  2  1  2  1  2  7  2  1  2  1  2  1 14
7     1  2  1  4  1  2  1  8  1  2  1  4  1  2  1 16
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A109004, A262343, A264906.

Table[GCD[2 n + 3 + k, k + 1], {n, 0, 8}, {k, 0, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 03 2016 *)

for (n=0, 20,for (k=0, 2*n+2, t=gcd(2*n+3+k, k+1); print1(t, ", ")))

T(n,k) = gcd(2*n+3+k, k+1), n >= 0, k = 0..2*n+1.

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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.)

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A266685 T(n,k) is the number of loops appearing in pattern of circular arc connecting two vertices of regular polygons. (See Comments.)

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Mathematica

PARI

Formula