A114296 First row of Modified Schroeder numbers for q=3 (A114292).

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

Author

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

See also A112833-A112844 and A114292-A114299.

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

Cf. A286761.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x or y 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 || yJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

a(n) ~ c * (3+2*sqrt(2))^n / n^(3/2), where c = 0.02741316010407391604887680145773... . - Vaclav Kotesovec, Sep 07 2014

Extensions

Corrected by Philippe Deléham, Sep 04 2006

Extended beyond a(10) by Alois P. Heinz, Apr 25 2013