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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 (list; graph; refs; listen; history; text; internal format)
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

See also A112833-A112844 and A114292-A114299.

Cf. A224776, A225041. - Alois P. Heinz, Apr 25 2013

Cf. A286761.

Sequence in context: A323229 A197158 A188314 * A121689 A192635 A009225

Adjacent sequences:  A114293 A114294 A114295 * A114297 A114298 A114299

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

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Last modified August 21 18:38 EDT 2019. Contains 326168 sequences. (Running on oeis4.)