A337318
Total number of nodes summed over all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.
Original entry on oeis.org
1, 2, 6, 16, 45, 131, 393, 1218, 3887, 12736, 42707, 146113, 508610, 1796848, 6428953, 23253209, 84893617, 312435085, 1157899672, 4317354453, 16183476500, 60947573729, 230481995102, 874810511970, 3331322503398, 12723257204883, 48722782351656, 187028551724723
Offset: 0
-
b:= proc(x, y) option remember; `if`(x=0, [1$2],
add(add((p-> p+[0, p[1]])(b(x-h, y-v)), h=1..
min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
-
b[x_, y_] := b[x, y] = If[x == 0, {1, 1},
Sum[Sum[Function[p, p + {0, p[[1]]}][b[x - h, y - v]], {h, 1,
Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)
A337863
Twice the total area of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.
Original entry on oeis.org
0, 0, 2, 8, 32, 123, 459, 1715, 6432, 24256, 91993, 350817, 1344559, 5176106, 20003683, 77567894, 301667089, 1176204195, 4596267116, 17995972170, 70581437130, 277244854010, 1090484851145, 4294324521033, 16929104620229, 66801975748076, 263827090563129
Offset: 0
-
b:= proc(x, y) option remember; `if`(x=0, [1, 0],
add(add((p-> p+[0, p[1]*h*(2*y-v)])(b(x-h, y-v)),
h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
-
b[x_, y_] := b[x, y] = If[x == 0, {1, 0},
Sum[Sum[Function[p, p + {0, p[[1]]*h*(2*y - v)}][b[x - h, y - v]],
{h, 1, Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)
A348202
Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.
Original entry on oeis.org
1, 1, 2, 4, 9, 22, 57, 155, 435, 1249, 3645, 10770, 32143, 96747, 293359, 895373, 2748803, 8483035, 26302248, 81896176, 255967640, 802790415, 2525691721, 7968972542, 25209580699, 79942927651, 254077293876, 809192984902, 2582113984084, 8254273128869
Offset: 0
-
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, add(b(x-1, y-j), j=[-4, -1, 0, 1])))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);
-
b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y - j], {j, {-4, -1, 0, 1}}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)
Showing 1-3 of 3 results.
