A069241 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| <= 2.

1, 1, 1, 3, 6, 10, 17, 28, 44, 68, 104, 157, 235, 350, 519, 767, 1131, 1665, 2448, 3596, 5279, 7746, 11362, 16662, 24430, 35815, 52501, 76956, 112797, 165325, 242309, 355135, 520490, 762830, 1117997, 1638520, 2401384, 3519416, 5157972, 7559393, 11078847

Offset: 0

Author

Don Knuth, Apr 13 2002

Keywords

• nonn
• easy

Comments

Equivalently, the number of bandwidth-at-most-2 arrangements of a straight line of n vertices.

Examples

			For example, the six Hamiltonian paths when n=4 are 1234, 1243, 1324, 1342, 2134, 3124.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Crossrefs

Cf. A003274, A000930, A092526, A263719, A302119.

Programs

Maple

a:= n-> (Matrix([[1,1,1,0,1]]). Matrix(5, (i,j)-> if i=j-1 then 1 elif j=1 then [3,-3,2,-2,1][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..50); # Alois P. Heinz, Sep 09 2008

Mathematica

a[0] = a[1] = a[2] = 1; a[3] = 3; a[4] = 6; a[n_] := a[n] = 3a[n-1] - 3a[n-2] + 2a[n-3] - 2a[n-4] + a[n-5]; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Feb 13 2015 *)
CoefficientList[Series[(3+x+x^2)/(1-x-x^3)-(2-x)/(1-x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{3,-3,2,-2,1},{1,1,1,3,6},60] (* Harvey P. Dale, Apr 07 2019 *)

Formula

a(n) = A003274(n)/2, n > 1.

a(n) = 3*s(n) + s(n-1) + s(n-2) - 2 - n, where s(n) = A000930(n).

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

Lim_{n->infinity} a(n+1)/a(n) = A092526 = 1/A263719. - Alois P. Heinz, Apr 15 2018