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%I #19 Mar 13 2023 11:57:23
%S 1,2,286,299,56653,56834,11006686,11009207,2135467321,2135502434,
%T 414272813758,414273302819,80366834417221,80366841228962,
%U 15590752217183806,15590752312059119,3024525571838019313,3024525573159461954,586742370303288400606,586742370321693722267,113824995314922590647741
%N Integer lengths of the sides of such regular hexagons that a polyline described in A356047 exists.
%C The length of the side of the hexagon is determined using a triangular grid depending on the number of links, which reduces to nontrivial solutions of the Pell equation x^2 - 3y^2 = 1 for even x.
%H Alexander M. Domashenko, <a href="https://elementy.ru/problems/2606/Zmeyka_v_shestiugolnike">Problem: Snake in a hexagon</a> (in Russian).
%H Alexander M. Domashenko, <a href="https://www.diofant.ru/problem/3996/">Problem 2211: Sixth hexagon</a> (in Russian).
%F a(n) = k(n)*sqrt((k(n)+1)^2/3 + 1)/4 for odd n,
%F a(n) = (k(n) + 1)*sqrt(k(n)^2/3 + 1)/4 for even n,
%F where k(n) = A356047(n).
%F Conjectures from _Chai Wah Wu_, Mar 13 2023: (Start)
%F a(n) = 208*a(n-2) - 2718*a(n-4) + 208*a(n-6) - a(n-8) for n > 8.
%F G.f.: x*(1+x)*(x^6+x^5+77*x^4-194*x^3+77*x^2+x+1) / ( (x^2+4*x+1) *(x^2-4*x+1) *(x^2-14*x+1) *(x^2+14*x+1) ). (End)
%Y Cf. A356047.
%K nonn,easy
%O 1,2
%A _Alexander M. Domashenko_, Oct 11 2022
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