This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A264906 #26 Sep 08 2022 08:46:14
%S A264906 25,36,49,64,81,100,121,72,169,196,225,256,289,324,361,100,441,484,
%T A264906 529,576,625,676,729,392,841,900,961,1024,1089,1156,1225,324,1369,
%U A264906 1444,1521,1600,1681,1764,1849,968,2025,2116,2209,2304,2401,2500,2601,676,2809
%N 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.)
%C A264906 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.)
%C A264906 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.
%H A264906 Kival Ngaokrajang, <a href="/A264906/a264906_1.pdf">Illustration of loop length L(n) for n = 5..12</a>
%F A264906 a(n) = n^2/gcd((n-4)/gcd(n-4,4),n); for n >= 5.
%e A264906 L(5) = 2*Pi - 1/25*Pi^3 + 1/7500*Pi^5 - 1/5625000*Pi^7 + 1/7875000000*Pi^9 - ...
%e A264906 L(6) = 2*Pi - 1/36*Pi^3 + 1/15552*Pi^5 - 1/16796160*Pi^7 + 1/33861058560*Pi^9 - ...
%e A264906 L(7) = 6*Pi - 3/49*Pi^3 + 1/9604*Pi^5 - 1/14117880*Pi^7 + 1/38739462720*Pi^9 - ...
%e A264906 L(8) = 2*Pi - 1/64*Pi^3 + 1/49152*Pi^5 - 1/94371840*Pi^7 + 1/338228674560*Pi^9 - ...
%e A264906 L(9) = 10*Pi - 5/81*Pi^3 + 5/78732*Pi^5 - 1/38263752*Pi^7 + 1/173564379072*Pi^9 - ...
%e A264906 L(10) = 6*Pi - 3/100*Pi^3 + 1/40000*Pi^5 - 1/120000000*Pi^7 + 1/672000000000*Pi^9 - ...
%e A264906 ...
%e A264906 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
%e A264906 T(n) = 2*(n-4) * Sum_{k>=0} (-1)^k * Pi^(2k+1) / ((2k)! * n^(2k)) for n=5..10.
%t A264906 Table[n^2/GCD[(n - 4)/GCD[n - 4, 4], n], {n, 5, 46}] (* _Michael De Vlieger_, Nov 28 2015 *)
%o A264906 (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)
%o A264906 (PARI) {for(n = 5, 100, a = n^2/gcd((n-4)/gcd(n-4,4), n); print1(a, ", "))} \\ a(n)
%o A264906 (Magma) [n^2 div Gcd((n-4) div Gcd(n-4,4),n): n in [5..60]]; // _Vincenzo Librandi_, Nov 29 2015
%Y A264906 Cf. A000265, A060819, A109375, A262343.
%K A264906 nonn,frac
%O A264906 5,1
%A A264906 _Kival Ngaokrajang_, Nov 28 2015
%E A264906 More terms from _Vincenzo Librandi_, Nov 29 2015