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A192371
Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).
1
1, 1, 3, 9, 25, 87, 307, 1113, 4149, 15605, 59201, 225999, 866449, 3333847, 12865335, 49769689, 192945411, 749396493, 2915432049, 11358771965, 44313108627, 173081422997, 676766482917, 2648843996031, 10376891445525, 40685535827325, 159641884780749, 626849029013919, 2463010645910537, 9683604464279235
OFFSET
0,3
LINKS
FORMULA
G.f.: sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)) where the function s satisfies (s^3-s-1)*(s-1)+x*s*(4-3*s) = 0. - Mark van Hoeij, Apr 17 2013
MAPLE
s := RootOf( (s^3-s-1)*(s-1)+x*s*(4-3*s), s);
ogf := sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)):
series(ogf, x=0, 30); # Mark van Hoeij, Apr 17 2013
# Alternative:
b:= proc(p) b(p):= `if`(p=[0$2], 1, `if`(min(p[])<0, 0,
add(b(p-l), l=[[1, 1], [0, 2], [2, 0], [0, 3], [3, 0]])))
end:
a:= n-> b([n$2]):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 18 2014
MATHEMATICA
b[p_List] := b[p] = If[p == {0, 0}, 1, If[Min[p] < 0, 0, Sum[b[p - l], {l, {{1, 1}, {0, 2}, {2, 0}, {3, 0}, {0, 3}}}]]]; a[n_] := b[{n, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
PROG
(PARI) /* same as in A092566 but use */
steps=[[1, 1], [2, 0], [0, 2], [3, 0], [0, 3]];
/* Joerg Arndt, Jun 30 2011 */
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jul 01 2011
STATUS
approved