OFFSET
0,3
COMMENTS
a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=x/2.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.
FORMULA
a(n) ~ c * (3+2*sqrt(2))^n / n^(3/2), where c = 0.02741316010407391604887680145773... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
The number of paths from (0,0) to (3,3) staying between the lines y=x and y=x/2 using steps of length (0,1), (1,0) and (1,1) is a(3)=5.
MAPLE
b:= proc(x, y) option remember; `if`(y>x or y<x/2, 0,
`if`(x=0, 1, b(x, y-1)+b(x-1, y)+b(x-1, y-1)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 25 2013
MATHEMATICA
b[x_, y_] := b[x, y] = If[y>x || y<x/2, 0, If[x == 0, 1, b[x, y-1] + b[x-1, y] + b[x-1, y-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
EXTENSIONS
Corrected by Philippe Deléham, Sep 04 2006
Extended beyond a(10) by Alois P. Heinz, Apr 25 2013
STATUS
approved