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A284461 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

%I A284461 #24 Oct 09 2019 13:35:16
%S A284461 1,5,111,5127,400593,47311677,7857786015,1745000283087,
%T A284461 499180661754849,178734707493557301,78294815164675006479,
%U A284461 41186656484051421462615,25619826402721039367943729,18600984174200732870460447213,15588291843672510150758754601407
%N A284461 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.
%H A284461 Alois P. Heinz, <a href="/A284461/b284461.txt">Table of n, a(n) for n = 0..224</a>
%H A284461 Alois P. Heinz, <a href="/A284461/a284461.gif">Animation of a(2)=111 walks</a>
%H A284461 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A284461 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A284461 a(n) = A284230(2n).
%F A284461 a(n) = Sum_{k=2n..n*(2n+3)} A284414(2n,k).
%p A284461 b:= proc(n) option remember; `if`(n<2, n+1,
%p A284461       (n+irem(n, 2))*b(n-1)+(n-1)*b(n-2))
%p A284461     end:
%p A284461 a:= n-> b(2*n):
%p A284461 seq(a(n), n=0..15);
%p A284461 # second Maple program:
%p A284461 a:= proc(n) option remember; `if`(n<2, 4*n+1,
%p A284461       ((2*n+1)^2-2)*a(n-1)-(4*n-6)*n*a(n-2))
%p A284461     end:
%p A284461 seq(a(n), n=0..15);
%t A284461 a[n_] := a[n] = If[n<2, 4n+1, ((2n+1)^2-2) a[n-1] - (4n-6) n a[n-2]];
%t A284461 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jun 19 2017, after 2nd Maple program *)
%Y A284461 Bisection of A284230 (even part).
%Y A284461 Cf. A284414, A285673.
%K A284461 nonn,walk
%O A284461 0,2
%A A284461 _Alois P. Heinz_, Mar 27 2017