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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 (list; graph; refs; listen; history; text; internal format)
 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: A223345 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

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Last modified July 21 19:13 EDT 2019. Contains 325199 sequences. (Running on oeis4.)