A357733 Integer lengths of the sides of such regular hexagons that a polyline described in A356047 exists.

1, 2, 286, 299, 56653, 56834, 11006686, 11009207, 2135467321, 2135502434, 414272813758, 414273302819, 80366834417221, 80366841228962, 15590752217183806, 15590752312059119, 3024525571838019313, 3024525573159461954, 586742370303288400606, 586742370321693722267, 113824995314922590647741

Offset: 1

Author

Alexander M. Domashenko, Oct 11 2022

Keywords

• nonn
• easy

Comments

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.

Links

• Alexander M. Domashenko, Problem: Snake in a hexagon (in Russian).
• Alexander M. Domashenko, Problem 2211: Sixth hexagon (in Russian).

Crossrefs

Cf. A356047.

Formula

a(n) = k(n)*sqrt((k(n)+1)^2/3 + 1)/4 for odd n,

a(n) = (k(n) + 1)*sqrt(k(n)^2/3 + 1)/4 for even n,

where k(n) = A356047(n).

Conjectures from Chai Wah Wu, Mar 13 2023: (Start)

a(n) = 208*a(n-2) - 2718*a(n-4) + 208*a(n-6) - a(n-8) for n > 8.

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)