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

1,2

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

Table of n, a(n) for n=1..21.

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

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

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)

Crossrefs

Cf. A356047.

Sequence in context: A307470 A182519 A279450 * A163276 A065498 A296591

Adjacent sequences: A357730 A357731 A357732 * A357734 A357735 A357736

Keyword

nonn,easy

Author

Alexander M. Domashenko, Oct 11 2022

Status

approved