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

0,4

Links

Table of n, a(n) for n=0..44.

Formula

T(n,k) = T(k,n).

Example

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).

Maple

T:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
     `if`(i^2+j^2<n^2+k^2, T(i, j), 0), j=k-1..k+1), i=n-1..n+1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..8);  # Alois P. Heinz, Sep 08 2021

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 *)

Crossrefs

Main diagonal gives A346539.

Column (or row) k=0 gives A347814.

Cf. A008288, A001850, A001263, A006318, A001006, A114486.

Sequence in context: A271178 A247950 A256844 * A166201 A248281 A111197

Adjacent sequences: A346535 A346536 A346537 * A346539 A346540 A346541

Keyword

nonn,tabl

Author

José María Grau Ribas, Jul 23 2021

Status

approved