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

1,2

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

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A025035, A005789.

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

Sequence in context: A369506 A066357 A276365 * A243921 A109092 A068416

Adjacent sequences: A185145 A185146 A185147 * A185149 A185150 A185151

Keyword

nonn

Author

Olivier Gérard, Feb 15 2011

Extensions

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

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

Status

approved