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A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).

%I A192365 #27 Apr 24 2025 02:13:25
%S A192365 1,3,22,165,1327,10950,92045,783579,6733966,58294401,507579829,
%T A192365 4440544722,39000863629,343677908223,3037104558574,26904952725061,
%U A192365 238854984979423,2124492829796598,18927927904130617,168888613467092895,1508973226894216106,13498652154574126523,120886709687492946083
%N A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).
%C A192365 Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x*y - (x*y)^2). - _Ilya Gutkovskiy_, Apr 23 2025
%H A192365 Alois P. Heinz, <a href="/A192365/b192365.txt">Table of n, a(n) for n = 0..1000</a>
%F A192365 G.f.: sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) ) where p4 = x^4+6*x^3+7*x^2-10*x+1. - _Mark van Hoeij_, Apr 16 2013
%p A192365 p4 := x^4+6*x^3+7*x^2-10*x+1;
%p A192365 ogf := sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) );
%p A192365 series(ogf, x=0, 30);  # _Mark van Hoeij_, Apr 16 2013
%p A192365 # second Maple program:
%p A192365 b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
%p A192365       `if`(max(x, y)=0, 1, add(b(x, y-j)+
%p A192365          b(x-j, y)+b(x-j, y-j), j=1..2)))
%p A192365     end:
%p A192365 a:= n-> b(n$2):
%p A192365 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 16 2017
%t A192365 b[x_, y_] := b[x, y] = If[Min[x, y] < 0, 0, If[Max[x, y] == 0, 1, Sum[b[x, y - j] + b[x - j, y] + b[x - j, y - j], {j, 1, 2}]]];
%t A192365 a[n_] := b[n, n];
%t A192365 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 23 2017, after _Alois P. Heinz_ *)
%o A192365 (PARI) /* same as in A092566 but use */
%o A192365 steps=[[0,1], [0,2], [1,0], [2,0], [1,1], [2,2]];
%o A192365 /* _Joerg Arndt_, Jun 30 2011 */
%K A192365 nonn
%O A192365 0,2
%A A192365 _Eric Werley_, Jun 29 2011
%E A192365 Terms > 507579829 from _Joerg Arndt_, Jun 30 2011