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.
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
Examples
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).
Crossrefs
Programs
-
Maple
T:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add( `if`(i^2+j^2 -
Mathematica
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 *)
Formula
T(n,k) = T(k,n).