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A185148
Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.
2
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
Column k=3 of A208615. - Alois P. Heinz, Feb 29 2012
Sequence in context: A369506 A066357 A276365 * A243921 A109092 A068416
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