A038577 Number of self-avoiding walks of length n from origin in strip Z X {0,1}.

1, 3, 6, 12, 20, 36, 58, 100, 160, 268, 430, 708, 1140, 1860, 3002, 4876, 7880, 12772, 20654, 33444, 54100, 87564, 141666, 229252, 370920, 600196, 971118, 1571340, 2542460, 4113828, 6656290, 10770148, 17426440, 28196620, 45623062, 73819716, 119442780

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• easy

Comments

For n >= 2, a(n) coincides with A110935. -Eric Rowland, Mar 09 2009

References

• J. Labelle, Self-avoiding walks and polyominoes in strips, Bull. ICA, 23 (1998), 88-98.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. Zeilberger, Self avoiding walks, the language of science and Fibonacci numbers, arXiv:math/9506214 [math.CO], 1995.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

Programs

Maple

f := n->if n mod 2 = 0 then 8*fibonacci(n)-n else 8*fibonacci(n)-4; fi;

Mathematica

Join[{1, 3}, LinearRecurrence[{1, 3, -2, -3, 1, 1}, {6, 12, 20, 36, 58, 100}, 40]] (* Jean-François Alcover, Jan 08 2019 *)

PARI

Vec((1 + 2*x - x^3 - x^4 + x^7) / ((1 - x)^2*(1 + x)^2*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, Nov 18 2017

Formula

G.f.: (1 + 2*x - x^3 - x^4 + x^7) / ((1 - x)^2*(1 + x)^2*(1 - x - x^2)).

From Colin Barker, Nov 18 2017: (Start)

a(n) = -2 + 2*(-1)^n - (8*(1/2-sqrt(5)/2)^n)/sqrt(5) + (8*(1/2+sqrt(5)/2)^n)/sqrt(5) - (1/2)*(1+(-1)^n)*n for n > 1.

a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 3*a(n-4) + a(n-5) + a(n-6) for n > 5.

(End)