A347811
Number A(n,k) of k-dimensional lattice walks from {n}^k to {0}^k using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 19, 25, 1, 1, 1, 323, 211075, 241, 1, 1, 1, 38716, 1322634996717, 2062017739, 2545, 1, 1, 1, 32253681, 16042961630858858915656, 29261778984922904560001, 32191353922714, 28203, 1, 1
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 19, 323, 38716, ...
1, 1, 25, 211075, 1322634996717, ...
1, 1, 241, 2062017739, 29261778984922904560001, ...
1, 1, 2545, 32191353922714, ...
1, 1, 28203, ...
...
-
s:= proc(n) option remember;
`if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
end:
b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
add(i^2, i=h) b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);
-
s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_List] := b[l] = With[{n = Length[l]}, If[l == Table[0, {n}], 1, Sum[With[{h = l+x}, If[h.h < l.l, b[Sort[h]], 0]], {x, s[n]}]]];
A[n_, k_] := b[Table[n, {k}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)
A346538
Table read by antidiagonals: T(n,k) is the number of paths in the Z X Z grid joining (0,0) and (n,k) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.
Original entry on oeis.org
1, 1, 1, 7, 3, 7, 29, 11, 11, 29, 173, 72, 25, 72, 173, 937, 382, 108, 108, 382, 937, 5527, 2295, 803, 241, 803, 2295, 5527, 32309, 13391, 4632, 1152, 1152, 4632, 13391, 32309, 193663, 80677, 29450, 9132, 2545, 9132, 29450, 80677, 193663
Offset: 0
Array begins:
1, 1, 7, 29, 173, 937, 5527, ...
1, 3, 11, 72, 382, 2295, 13391, ...
7, 11, 25, 108, 803, 4632, 29450, ...
29, 72, 108, 241, 1152, 9132, 56043, ...
173, 382, 803, 1152, 2545, 12829, 106207, ...
937, 2295, 4632, 9132, 12829, 28203, 147239, ...
5527, 13391, 29450, 56043, 106207, 147239, 322681, ...
...
T(6,4) = T(5,3) + T(5,4) + T(5,5) + T(6,3) = 9132 + 12829 + 28203 + 56043 =106207.
T(7,5) = T(6,4) + T(6,5) + T(6,6) + T(7,4).
T(7,6) = T(6,6) + T(7,5) + T(6,5).
T(0,5) = T(-1,4) + T(0,4) + T(1,4).
-
T:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
`if`(i^2+j^2Alois P. Heinz, Sep 08 2021
-
rodean[{m_, n_}] := Select[ Complement[ Flatten[Table[{m, n} + {s, t}, {s, -1, 1}, {t, -1, 1}], 1] // Union, {{m, n}}], #[[1]]^2 + #[[2]]^2 < m^2 + n^2 &];
$RecursionLimit = 10^6; Clear[T]; T[{0, 0}] = 1;
T[{m_, n_}] := T[{m, n}] = Sum[T[rodean[{m, n}][[i]]],{i,Length[rodean[{m,n}]]}] ;
Table[T[{k, n - k}], {n, 0, 12}, {k, 0, n}] // Flatten
(* Second program: *)
T[n_, k_] := T[n, k] = If[{n, k} == {0, 0}, 1, Sum[Sum[If[i^2 + j^2 < n^2 + k^2, T[i, j], 0], {j, k - 1, k + 1}], {i, n - 1, n + 1}]];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)
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