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A284230 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), 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 A284230 #31 Oct 09 2019 13:34:39
%S A284230 1,2,5,24,111,762,5127,45588,400593,4370634,47311677,611446464,
%T A284230 7857786015,117346361778,1745000283087,29562853594284,499180661754849,
%U A284230 9458257569095826,178734707493557301,3744942786114870888,78294815164675006479,1797384789345147560298
%N A284230 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), 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 A284230 Alois P. Heinz, <a href="/A284230/b284230.txt">Table of n, a(n) for n = 0..448</a>
%H A284230 Alois P. Heinz, <a href="/A284230/a284230.gif">Animation of a(4)=111 walks</a>
%H A284230 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A284230 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A284230 a(n) ~ c * n^(n+2) / exp(n), where c = 0.7741273379869056907732932906458364317717498069987762339667734187318... - _Vaclav Kotesovec_, Mar 27 2017
%F A284230 Conjecture: a(n) -a(n-1) +(-n^2-n+3)*a(n-2) +(-n+2)*a(n-3) +(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Apr 09 2017
%e A284230 a(0) = 1: [(0,0)].
%e A284230 a(1) = 2: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
%e A284230 a(2) = 5: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(1,1),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)].
%p A284230 a:= proc(n) option remember; `if`(n<2, n+1,
%p A284230       (n+irem(n, 2))*a(n-1)+(n-1)*a(n-2))
%p A284230     end:
%p A284230 seq(a(n), n=0..25);
%t A284230 a[n_]:=If[n<2, n + 1, (n + Mod[n,2]) * a[n - 1] + (n - 1) a[n - 2]]; Table[a[n], {n, 0, 25}] (* _Indranil Ghosh_, Mar 27 2017 *)
%Y A284230 Row sums of A284414.
%Y A284230 Bisection (even part) gives A284461.
%Y A284230 Cf. A001900, A277358, A284231, A285673.
%K A284230 nonn,walk
%O A284230 0,2
%A A284230 _Alois P. Heinz_, Mar 23 2017