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A264906
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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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2
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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
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OFFSET
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5,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = n^2/gcd((n-4)/gcd(n-4,4),n); for n >= 5.
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EXAMPLE
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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.
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MATHEMATICA
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Table[n^2/GCD[(n - 4)/GCD[n - 4, 4], n], {n, 5, 46}] (* Michael De Vlieger, Nov 28 2015 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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