A328300
Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (0,k,n-k), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 26, 15, 1, 1, 31, 82, 82, 31, 1, 1, 63, 237, 343, 237, 63, 1, 1, 127, 651, 1257, 1257, 651, 127, 1, 1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1, 1, 511, 4494, 13669, 22411, 22411, 13669, 4494, 511, 1, 1, 1023, 11485, 42279, 83680, 103730, 83680, 42279, 11485, 1023, 1
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 26, 15, 1;
1, 31, 82, 82, 31, 1;
1, 63, 237, 343, 237, 63, 1;
1, 127, 651, 1257, 1257, 651, 127, 1;
1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1;
...
-
b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
T:= (n, k)-> b(sort([0, k, n-k])):
seq(seq(T(n, k), k=0..n), n=0..12);
-
b[l_List] := b[l] = If[l[[-1]] == 0, 1, Function[r, Sum[Sum[Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {k, r}], {j, r}], {i, r}]][{-1, 0, 1}]];
T[n_, k_] := b[Sort[{0, k, n - k}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 10 2020, after Maple *)
A328296
Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,y,z), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
Original entry on oeis.org
1, 2, 5, 16, 58, 228, 945, 4072, 18078, 82172, 380666, 1791138, 8537912, 41146988, 200169891, 981705400, 4848820372, 24098703860, 120433164750, 604831645542, 3050979757728, 15451575335362, 78536766518038, 400497435480332, 2048473941706016, 10506489209380466
Offset: 0
-
b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
a:= n-> add(b(sort([0, j, n-j])), j=0..n):
seq(a(n), n=0..29);
-
b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := Sum[b[Sort[{0, j, n - j}]], {j, 0, n }];
a /@ Range[0, 29] (* Jean-François Alcover, May 12 2020, after Maple *)
A328299
Number of n-step walks on cubic lattice starting at (0,0,0), ending at (floor(n/3), floor((n+1)/3), floor((n+2)/3)), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
Original entry on oeis.org
1, 1, 3, 12, 41, 179, 909, 3968, 19680, 106368, 516905, 2717631, 15139485, 77813569, 422589823, 2395441908, 12734635078, 70577595746, 404540380566, 2199035619696, 12356298623126, 71368686011040, 394076753535029, 2236273925952447, 12988459939106601
Offset: 0
a(2) = 3: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)].
-
b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
a:= n-> b([floor((n+i)/3)$i=0..2]):
seq(a(n), n=0..24);
-
b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := b[Table[Floor[(n+i)/3], {i, 0, 2}]];
a /@ Range[0, 24] (* Jean-François Alcover, May 12 2020, after Maple *)
A328295
Number of n-step walks on cubic lattice starting at (0,0,0), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
Original entry on oeis.org
1, 3, 12, 57, 294, 1590, 8856, 50301, 289590, 1683306, 9853320, 57977922, 342494118, 2029350972, 12052664868, 71715479535, 427347761010, 2549540104944, 15224944518084, 90988367614254, 544115710748898, 3255541325220204, 19486893225315138, 116685749052336714
Offset: 0
-
b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
a:= n-> add(add(b(sort([i, j, n-i-j])), j=0..n-i), i=0..n):
seq(a(n), n=0..23);
-
b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][{-1, 0, 1}]];
a[n_] := Sum[Sum[b[Sort[{i, j, n - i - j}]], {j, 0, n - i}], {i, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 13 2020, after Maple *)
A328427
Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (n,y,z), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
Original entry on oeis.org
1, 6, 93, 2040, 52086, 1443898, 42073956, 1266666928, 39005460010, 1220921798726, 38687295919777, 1237507863154364, 39880747868561408, 1292960017095690800, 42125333404129589074, 1378096053159532505284, 45239194620411006084462, 1489461895371375668384236
Offset: 0
-
b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
a:= n-> add(b(sort([n, j, n-j])), j=0..n):
seq(a(n), n=0..23);
-
b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][{-1, 0, 1}]];
a[n_] := Sum[b[Sort[{n, j, n - j}]], {j, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 13 2020, after Maple *)
Showing 1-5 of 5 results.