A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

1, 6, 53, 587, 7572, 109027, 1705249, 28440320, 499208817, 9134237407, 172976239886, 3371587949969, 67351686970929, 1374179898145980, 28557595591148315, 603118526483125869, 12920388129877471030, 280324904918707937001, 6151595155000424589327, 136384555249451824930126

Offset: 1

Author

Olivier Gérard, Feb 15 2011

Keywords

• nonn

Comments

a(n) counts a subset of A025035(n).

a(n) counts a more general set than A005789(n).

a(n) is also the number of (3*n-1)-step walks on 3-dimensional cubic lattice from (1,0,0) to (n,n,n) with steps in {(1,0,0), (0,1,0), (0,0,1)} such that for each point (x,y,z) we have x<=y<=z or x>=y>=z. - Alois P. Heinz, Feb 29 2012

Examples

			For n = 2 the a(2) = 6 arrangements are:
+---+  +---+  +---+  +---+  +---+  +---+
|1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|
|2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|
|3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|
+---+  +---+  +---+  +---+  +---+  +---+
Only the second of these arrangements is not counted by A005789(2).
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..700 (terms 0..200 from Alois P. Heinz)

Crossrefs

Cf. A025035, A005789.

Column k=3 of A208615. - Alois P. Heinz, Feb 29 2012

Programs

Maple

b:= proc(x, y, z) option remember;
      `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+
      `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))
    end:
a:= n-> b(n-1, n$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 29 2012

Mathematica

b[x_, y_, z_] := b[x, y, z] = If[x == z, If[x == 0, 1, 2*b[x - 1, y, z]], If[x > 0, b[x - 1, y, z], 0] + If[y > x, b[x, y - 1, z], 0] + If[z > y, b[x, y, z - 1], 0]];
a[n_] := b[n - 1, n, n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 12 2017, after Alois P. Heinz *)

Formula

a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - Vaclav Kotesovec, Sep 03 2014, updated Sep 07 2016

Extensions

More terms and example from Alois P. Heinz, Feb 22 2011

Extended beyond a(8) by Alois P. Heinz, Feb 22 2012