A333574 - OEIS

%I #42 Sep 09 2024 15:34:51

%S 1,2,4,6,10,14,20,26,34,42,52,62,74,86,100,114,130,146,164,182,202,

%T 222,244,266,290,314,340,366,394,422,452,482,514,546,580,614,650,686,

%U 724,762,802,842,884,926,970,1014,1060,1106,1154,1202,1252,1302,1354,1406,1460

%N Number of Hamiltonian paths in the n X 2 grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.

%C Conjecture: Numbers k such that A339399(k) = A103128(k). - _Wesley Ivan Hurt_, Nov 19 2021

%H Colin Barker, <a href="/A333574/b333574.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

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

%F From _Colin Barker_, Mar 27 2020: (Start)

%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5.

%F a(n) = (9 + (-1)^(1+n) - 4*n + 2*n^2) / 4 for n>1. (End)

%F E.g.f.: ((4 - x + x^2)*cosh(x) + (5 - x + x^2)*sinh(x) - 2*(2 + x))/2. - _Stefano Spezia_, Jun 14 2023

%e a(1) = 1;

%e    +--+

%e a(2) = 2;

%e    +  +   *--*

%e    |  |   |  |

%e    *--*   +  +

%e a(3) = 4;

%e    +  +   +--*   *--+   *--*

%e    |  |      |   |      |  |

%e    *  *   *--*   *--*   *  *

%e    |  |   |         |   |  |

%e    *--*   *--+   +--*   +  +

%o (PARI) N=66; x='x+O('x^N); Vec(x*(1+2*x*(1-x^2+x^3)/((1+x)*(1-x)^3)))

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o import graphillion.tutorial as tl

%o def A(start, goal, n, k):

%o     universe = tl.grid(n - 1, k - 1)

%o     GraphSet.set_universe(universe)

%o     paths = GraphSet.paths(start, goal, is_hamilton=True)

%o     return paths.len()

%o def A333571(n, k):

%o     if n == 1: return 1

%o     s = 0

%o     for i in range(1, n + 1):

%o         for j in range(k * n - n + 1, k * n + 1):

%o             s += A(i, j, k, n)

%o     return s

%o def A333574(n):

%o     return A333571(n, 2)

%o print([A333574(n) for n in range(1, 25)])

%Y Column k=2 of A333571.

%Y Cf. A333510.

%Y Cf. A103128, A339399.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Mar 27 2020