|
|
A038577
|
|
Number of self-avoiding walks of length n from origin in strip Z X {0,1}.
|
|
7
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
|
|
REFERENCES
|
J. Labelle, Self-avoiding walks and polyominoes in strips, Bull. ICA, 23 (1998), 88-98.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1 + 2*x - x^3 - x^4 + x^7) / ((1 - x)^2*(1 + x)^2*(1 - x - x^2)).
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)
|
|
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 *)
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,walk,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|