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A346540
Number A(n,k) of walks on square lattice from (n,k) to (0,0) using steps that decrease the Euclidean distance to the origin and increase the Euclidean distance to (n,k) and that change each coordinate by at most 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
3
1, 1, 1, 3, 3, 3, 7, 5, 5, 7, 19, 24, 13, 24, 19, 51, 81, 25, 25, 81, 51, 141, 298, 173, 63, 173, 298, 141, 393, 1070, 739, 129, 129, 739, 1070, 393, 1107, 3868, 3423, 1210, 321, 1210, 3423, 3868, 1107, 3139, 13960, 15363, 6273, 681, 681, 6273, 15363, 13960, 3139
OFFSET
0,4
COMMENTS
Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.
FORMULA
A(n,k) = A(k,n).
EXAMPLE
Square array A(n,k) begins:
1, 1, 3, 7, 19, 51, 141, 393, ...
1, 3, 5, 24, 81, 298, 1070, 3868, ...
3, 5, 13, 25, 173, 739, 3423, 15363, ...
7, 24, 25, 63, 129, 1210, 6273, 34318, ...
19, 81, 173, 129, 321, 681, 8371, 51727, ...
51, 298, 739, 1210, 681, 1683, 3653, 57644, ...
141, 1070, 3423, 6273, 8371, 3653, 8989, 19825, ...
393, 3868, 15363, 34318, 51727, 57644, 19825, 48639, ...
...
MAPLE
s:= proc(n) option remember;
`if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
end:
b:= proc(l, v) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
add(i^2, i=h)<add(i^2, i=l) and add(i^2, i=v-h)>add(i^2, i=v-l)
, b(h, v), 0))(l+x), x=s(n))))(nops(l))
end:
A:= (n, k)-> b(sort([n, k])$2):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
s[n_] := s[n] = If[n == 0, {{}}, Sequence @@
Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_, v_] := b[l, v] = With[{n = Length[l]},
If[l == Table[0, {n}], 1, Sum[With[{h = l+x},
If[h.h<l.l && (v-h).(v-h)>(v-l).(v-l), b[h, v], 0]], {x, s[n]}]]];
A[n_, k_] := b[Sort[{n, k}], Sort[{n, k}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)
CROSSREFS
Columns (or rows) k=0-1 give: A002426, A347947.
Main diagonal gives A001850.
A(2n,n) gives A346541.
A(n-1,n) gives A002002 for n >= 1.
Sequence in context: A294505 A242016 A084038 * A326930 A095108 A132197
KEYWORD
nonn,tabl,walk
AUTHOR
Alois P. Heinz, Sep 16 2021
STATUS
approved