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A114296
First row of Modified Schroeder numbers for q=3 (A114292).
4
1, 1, 2, 5, 16, 57, 224, 934, 4092, 18581, 86888, 415856, 2029160, 10061161, 50568680, 257129888, 1320619176, 6842177174, 35722456976, 187772944964, 992991472328, 5279633960181, 28208037066528, 151373637844440, 815568695756496, 4410124252008112
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
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
Cf. A224776, A225041. - Alois P. Heinz, Apr 25 2013
Cf. A286761.
Sequence in context: A323229 A197158 A188314 * A121689 A357580 A192635
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