A185148 - OEIS

%I #43 Nov 12 2017 03:02:50

%S 1,6,53,587,7572,109027,1705249,28440320,499208817,9134237407,

%T 172976239886,3371587949969,67351686970929,1374179898145980,

%U 28557595591148315,603118526483125869,12920388129877471030,280324904918707937001,6151595155000424589327,136384555249451824930126

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

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

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

%C 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

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A185148/b185148.txt">Table of n, a(n) for n = 1..700</a> (terms 0..200 from Alois P. Heinz)

%F a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - _Vaclav Kotesovec_, Sep 03 2014, updated Sep 07 2016

%e For n = 2 the a(2) = 6 arrangements are:

%e +---+  +---+  +---+  +---+  +---+  +---+

%e |1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|

%e |2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|

%e |3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|

%e +---+  +---+  +---+  +---+  +---+  +---+

%e Only the second of these arrangements is not counted by A005789(2).

%p b:= proc(x, y, z) option remember;

%p       `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+

%p       `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))

%p     end:

%p a:= n-> b(n-1, n$2):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Feb 29 2012

%t 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]];

%t a[n_] := b[n - 1, n, n];

%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Nov 12 2017, after _Alois P. Heinz_ *)

%Y Cf. A025035, A005789.

%Y Column k=3 of A208615. - _Alois P. Heinz_, Feb 29 2012

%K nonn

%O 1,2

%A _Olivier Gérard_, Feb 15 2011

%E More terms and example from _Alois P. Heinz_, Feb 22 2011

%E Extended beyond a(8) by _Alois P. Heinz_, Feb 22 2012