Jean-Pierre Levrel - cp's OEIS frontend

User: Jean-Pierre Levrel

Jean-Pierre Levrel has authored 1 sequences.

A247150 Number of paths from (0,0,0) to (n,n,n) avoiding 3 or more consecutive right steps, 3 or more consecutive up steps, and 3 or more consecutive away steps.

1, 6, 90, 1314, 21084, 353772, 6128208, 108606408, 1958248980, 35787633828, 661145207064, 12322983505860, 231395387482470, 4372431546366636, 83068148270734740, 1585548331063624992, 30388252830928088010, 584527926996090202428, 11279880522021539956860

Offset: 0

Jean-Pierre Levrel, Nov 21 2014

nonn

This is a generalization of A177790 from 2D to 3D.

a(n) is also the number of ternary vectors (symbols 0, 1, and 2, for example) that can be composed with 3n elements (same number of each of the symbols) where each symbol cannot be repeated more than twice consecutively. For example, 0,2,1,0,2,2,1,0,1 is allowed, but 0,2,1,1,1,2,2,0,0 is prohibited because the symbol 1 is repeated 3 times.

			For n=1 the 6 paths are (000>001>011>111), (000>001>101>111), (000>010>011>111), (000>010>110>111), (000>100>101>111), (000>100>110>111).

Alois P. Heinz, Table of n, a(n) for n = 0..250
M. Erickson, S. Fernando, K. Tran, Enumerating rook and queen paths, Bulletin of the Institute for Combinatorics and Its Applications, Volume 60 (2010), 37-48.

Cf. A177790.

f:= proc(p,q,r) option remember;
  if p`))) fi;
  if r < 0 then return 0 fi;
   procname(p-1,q-1,r)+procname(p-1,q-2,r)+procname(p-2,q-1,r)+procname(p-2,q-2,r)+2*procname(p-1,q-1,r-1)+procname(p,q-2,r-1)+2*procname(p-1,q-2,r-1)+procname(p-1,q,r-1)+2*procname(p-2,q-1,r-1)+procname(p-2,q,r-1)+2*procname(p-2,q-2,r-1)+procname(p,q-1,r-1)+2*procname(p-2,q-2,r-2)+procname(p,q-1,r-2)+2*procname(p-1,q-1,r-2)+procname(p,q-2,r-2)+2*procname(p-1,q-2,r-2)+procname(p-1,q,r-2)+2*procname(p-2,q-1,r-2)+procname(p-2,q,r-2)
end proc:
f(0,0,0) := 1: f(1,0,0) := 1:
f(1,1,0) := 2: f(1,1,1) := 6:
f(2,0,0) := 1: f(2,1,0) := 3:
f(2,1,1) := 12: f(2,2,0) := 6:
f(2,2,1) := 30: f(2,2,2) := 90:
seq(f(n,n,n), n=0..30); # Robert Israel, Nov 26 2014
# second Maple program:
b:= proc(i, j, k, t) option remember; `if`(max(i, j, k)=0, 1,
      `if`(j>0, b(j-1, `if`(i0, b(k-1, `if`(i0 and t>0, b(i-1, j, k, t-1), 0))
    end:
a:= n-> b(n$3, 2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 26 2014

(* Very slow *) a[0] = 1; a[n_] := SeriesCoefficient[((1+x+x^2)*(1+y+y^2)*(1+z+z^2)/(1-x*y*(1+x)*(1+y) - x*z*(1+x)*(1+ z) - y*z*(1+y)*(1+z) - 2*x*y*z*(1+x)*(1+y)*(1+z))), {x, 0, n}, {y, 0, n}, {z, 0, n}]; Table[Print[an = a[n]]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 26 2014 *)
b[i_, j_, k_, t_] := b[i, j, k, t] = If[Max[i, j, k] == 0, 1, If[j>0, If[i0, If[i0 && t>0, b[i-1, j, k, t-1], 0]]; a[n_] := b[n, n, n, 2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 27 2014, after Alois P. Heinz *)

a(n) = [x^n y^n z^n] ((1+x+x^2)*(1+y+y^2)*(1+z+z^2)/(1-x*y*(1+x)*(1+y)-x*z*(1+x)*(1+z)-y*z*(1+y)*(1+z)-2*x*y*z*(1+x)*(1+y)*(1+z))).
Recurrence (20 terms):
a(p,q,r) = a(p-1,q-1,r) +a(p-1,q-2,r) +a(p-2,q-1,r) +a(p-2,q-2,r) +2*a(p-1,q-1,r-1) +a(p,q-2,r-1) +2*a(p-1,q-2,r-1) +a(p-1,q,r-1) +2*a(p-2,q-1,r-1) +a(p-2,q,r-1) +2*a(p-2,q-2,r-1) +a(p,q-1,r-1) +2*a(p-2,q-2,r-2) +a(p,q-1,r-2) +2*a(p-1,q-1,r-2) +a(p,q-2,r-2) +2*a(p-1,q-2,r-2) +a(p-1,q,r-2) +2*a(p-2,q-1,r-2) +a(p-2,q,r-2), for (p,q,r) > 2.
a(p,q,r) = 0 when p or q or r is negative.
Initial conditions: a(0,0,0) = 1, a(1,0,0) = 1, a(1,1,0) = 2, a(1,1,1) = 6, a(2,0,0) = 1, a(2,1,0) = 3, a(2,1,1) = 12, a(2,2,0) = 6, a(2,2,1) = 30, a(2,2,2) = 90.
Symmetry: a(p,q,r) = a(p,r,q) = a(q,p,r) = a(q,r,p) = a(r,p,q) = a(r,q,p).

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User: Jean-Pierre Levrel

A247150 Number of paths from (0,0,0) to (n,n,n) avoiding 3 or more consecutive right steps, 3 or more consecutive up steps, and 3 or more consecutive away steps.

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