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 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
 1, 6, 90, 1314, 21084, 353772, 6128208, 108606408, 1958248980, 35787633828, 661145207064, 12322983505860, 231395387482470, 4372431546366636, 83068148270734740, 1585548331063624992, 30388252830928088010, 584527926996090202428, 11279880522021539956860 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS 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 FORMULA 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). EXAMPLE 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). MAPLE 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 MATHEMATICA (* 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 *) CROSSREFS Cf. A177790. Sequence in context: A091800 A317487 A037959 * A201073 A006480 A138462 Adjacent sequences:  A247147 A247148 A247149 * A247151 A247152 A247153 KEYWORD nonn AUTHOR Jean-Pierre Levrel, Nov 21 2014 STATUS approved

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Last modified October 17 14:54 EDT 2018. Contains 316282 sequences. (Running on oeis4.)