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.
Original entry on oeis.org
1, 2, 5, 24, 111, 762, 5127, 45588, 400593, 4370634, 47311677, 611446464, 7857786015, 117346361778, 1745000283087, 29562853594284, 499180661754849, 9458257569095826, 178734707493557301, 3744942786114870888, 78294815164675006479, 1797384789345147560298
Offset: 0
a(0) = 1: [(0,0)].
a(1) = 2: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
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)].
Bisection (even part) gives
A284461.
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a:= proc(n) option remember; `if`(n<2, n+1,
(n+irem(n, 2))*a(n-1)+(n-1)*a(n-2))
end:
seq(a(n), n=0..25);
-
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 *)
A284231
Total number of nodes summed over all 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.
Original entry on oeis.org
1, 5, 21, 152, 975, 8835, 75499, 830180, 8819417, 114384573, 1450018173, 21689509992, 319180726887, 5411092531323, 90615453774771, 1717272516535812, 32234085990345105, 675335923050095253, 14040521125141683717, 322252846702242056280, 7349647183279936080543
Offset: 0
a(0) = 1: [(0,0)].
a(1) = 5: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
a(2) = 21: [(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)].
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.
Original entry on oeis.org
1, 5, 111, 5127, 400593, 47311677, 7857786015, 1745000283087, 499180661754849, 178734707493557301, 78294815164675006479, 41186656484051421462615, 25619826402721039367943729, 18600984174200732870460447213, 15588291843672510150758754601407
Offset: 0
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b:= proc(n) option remember; `if`(n<2, n+1,
(n+irem(n, 2))*b(n-1)+(n-1)*b(n-2))
end:
a:= n-> b(2*n):
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n+1,
((2*n+1)^2-2)*a(n-1)-(4*n-6)*n*a(n-2))
end:
seq(a(n), n=0..15);
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a[n_] := a[n] = If[n<2, 4n+1, ((2n+1)^2-2) a[n-1] - (4n-6) n a[n-2]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 19 2017, after 2nd Maple program *)
A277424
Total number of nodes summed over all 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).
Original entry on oeis.org
1, 5, 39, 379, 4457, 61503, 974107, 17412317, 346662981, 7605810685, 182298744203, 4738700778123, 132767583248917, 3988244997744743, 127859570155253607, 4357113615504651565, 157266354405499307369, 5993377455733610208885, 240479249123008267155343
Offset: 0
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b:= proc(x, y, t) option remember; `if`(x<0 or y<0, 0,
`if`(x=0 and y=0, [1$2], (p-> p+ [0, p[1]])(
b(x-1, y, 0)+ b(x, y-1, 0)+ b(x-1, y-1, 0)+
`if`(t<>2, b(x+1, y-1, 1), 0)+
`if`(t<>1, b(x-1, y+1, 2), 0))))
end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..25);
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b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, 0, If[x == 0 && y == 0, {1, 1}, Function[p, p + {0, p[[1]]}][b[x - 1, y, 0] + b[x, y - 1, 0] + b[x - 1, y - 1, 0] + If[t != 2, b[x + 1, y - 1, 1], 0] + If[t != 1, b[x - 1, y + 1, 2], 0]]]];
a[n_] := b[n, 0, 0][[2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 19 2017, translated from Maple *)
A277756
Total number of nodes summed over all self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.
Original entry on oeis.org
1, 5, 32, 224, 1723, 14569, 135286, 1375882, 15263414, 183817326, 2391291386, 33443618930, 500611975023, 7988044467121, 135376576319870, 2428721569276698, 45988428905194350, 916607431346170686, 19182997480530342168, 420606731490047403144
Offset: 0
-
b:= proc(x, y, t) option remember; `if`(x<0 or y<0, 0,
`if`(x=0 and y=0, [1$2], (p-> p+[0, p[1]])(
`if`(y=x, b(x-1, y-1, 0), 0)+
`if`(y>x+1 and t<>2, b(x+1, y-1, 1), 0)+
`if`(y>=x and t<>1, b(x-1, y+1, 2), 0))))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..25);
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b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, {0, 0}, If[x == 0 && y == 0, {1, 1}, # + {0, #[[1]]}&[If[y < x, b[x-1, y, 0], 0] + If[y <= x, b[x, y-1, 0], 0] + If[y >= x, b[x-1, y-1, 0], 0] + If[y > x+1 && t != 2, b[x+1, y-1, 1], 0] + If[y >= x && t != 1, b[x-1, y+1, 2], 0]]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 25] (* Jean-François Alcover, Oct 19 2019, after Alois P. Heinz *)
Showing 1-5 of 5 results.
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