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A357733
Integer lengths of the sides of such regular hexagons that a polyline described in A356047 exists.
1
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved