A356047 - OEIS

%I #38 Oct 19 2024 12:05:40

%S 2,3,44,45,626,627,8732,8733,121634,121635,1694156,1694157,23596562,

%T 23596563,328657724,328657725,4577611586,4577611587,63757904492,

%U 63757904493,888033051314,888033051315,12368704813916,12368704813917,172273834343522,172273834343523,2399464975995404,2399464975995405,33420235829592146,33420235829592147

%N The number of links of a polyline that connects the midpoints of opposite sides of the n-th regular integer hexagon and has the following properties: the first link is 1; each subsequent one is 1 more than the previous one; the angle between adjacent links is equal to Pi/3; links of the same parity are parallel.

%C The number of links a(n) is determined using a triangular grid from the dependence of the integer side of the hexagon on a(n), which reduces to nontrivial solutions to the Pell equation x^2 - 3y^2 = 1 for even x.

%C In the definition, "n-th regular integer hexagon" means the n-th integer-sided regular hexagon such that the polyline described in the name is possible. These hexagons have sides A357733(n). - _Andrey Zabolotskiy_, Jul 30 2022

%H Paolo Xausa, <a href="/A356047/b356047.txt">Table of n, a(n) for n = 1..1500</a>

%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).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,15,0,-15,0,1)

%F For odd n: a(n) = 3*y(n) - 1 from the nontrivial solution of the equation x^2 - 3y^2 = 1;

%F for even n: a(n) = 3*y(n-1) from the nontrivial solution of the equation x^2 - 3y^2 = 1.

%F Here y(n) = A001353(n). - _Andrey Zabolotskiy_, Oct 16 2022

%F From _Chai Wah Wu_, Mar 13 2023: (Start)

%F a(n) = 15*a(n-2) - 15*a(n-4) + a(n-6) for n > 6.

%F G.f.:  x*(-2-3*x-14*x^2+4*x^4+3*x^5) / ( (x-1)*(1+x)*(x^2-4*x+1)*(x^2+4*x+1) ). (End)

%e a(1) = 2, since the first nontrivial pair (2;1) of the Pell equation x^2 - 3y^2 = 1 determines a(1) = 3*y(1) - 1 = 3*1 - 1 = 2 and in a hexagon with side 1 a broken line of two links connects the midpoints of its opposite sides.

%e a(2) = 3, since the first nontrivial pair (2;1) of the Pell equation x^2 - 3y^2 = 1 determines a(2) = 3*y(2 -1) = 3 and in a hexagon with side 2 a broken line of three links connects the midpoints of its opposite sides.

%e a(3) = 44, since the third nontrivial pair (26;15) of the Pell equation x^2 - 3y^2 = 1 determines a(3) = 3*y(3) - 1 = 3*15 - 1 = 44.

%e a(4) = 45, since the third nontrivial pair (26;15) of the Pell equation x^2 - 3y^2 = 1 determines a(4) = 3*y(4 -1) = 3*15 = 45.

%t LinearRecurrence[{0, 15, 0, -15, 0, 1}, {2, 3, 44, 45, 626, 627}, 30] (* _Paolo Xausa_, Oct 03 2024 *)

%Y Cf. A001353, A357733.

%K nonn,easy

%O 1,1

%A _Alexander M. Domashenko_, Jul 24 2022