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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.

%I A346538 #39 Nov 03 2021 08:38:44
%S A346538 1,1,1,7,3,7,29,11,11,29,173,72,25,72,173,937,382,108,108,382,937,
%T A346538 5527,2295,803,241,803,2295,5527,32309,13391,4632,1152,1152,4632,
%U A346538 13391,32309,193663,80677,29450,9132,2545,9132,29450,80677,193663
%N 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.
%F A346538 T(n,k) = T(k,n).
%e A346538 Array begins:
%e A346538      1,     1,     7,    29,    173,    937,   5527, ...
%e A346538      1,     3,    11,    72,    382,   2295,  13391, ...
%e A346538      7,    11,    25,   108,    803,   4632,  29450, ...
%e A346538     29,    72,   108,   241,   1152,   9132,  56043, ...
%e A346538    173,   382,   803,  1152,   2545,  12829, 106207, ...
%e A346538    937,  2295,  4632,  9132,  12829,  28203, 147239, ...
%e A346538   5527, 13391, 29450, 56043, 106207, 147239, 322681, ...
%e A346538   ...
%e A346538 T(6,4) = T(5,3) + T(5,4) + T(5,5) + T(6,3) = 9132 + 12829 + 28203 + 56043 =106207.
%e A346538 T(7,5) = T(6,4) + T(6,5) + T(6,6) + T(7,4).
%e A346538 T(7,6) = T(6,6) + T(7,5) + T(6,5).
%e A346538 T(0,5) = T(-1,4) + T(0,4) + T(1,4).
%p A346538 T:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
%p A346538      `if`(i^2+j^2<n^2+k^2, T(i, j), 0), j=k-1..k+1), i=n-1..n+1))
%p A346538     end:
%p A346538 seq(seq(T(n, d-n), n=0..d), d=0..8);  # _Alois P. Heinz_, Sep 08 2021
%t A346538 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 &];
%t A346538 $RecursionLimit = 10^6; Clear[T]; T[{0, 0}] = 1;
%t A346538 T[{m_, n_}] := T[{m, n}] = Sum[T[rodean[{m, n}][[i]]],{i,Length[rodean[{m,n}]]}] ;
%t A346538 Table[T[{k, n - k}], {n, 0, 12}, {k, 0, n}] // Flatten
%t A346538 (* Second program: *)
%t A346538 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}]];
%t A346538 Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* _Jean-François Alcover_, Nov 03 2021, after _Alois P. Heinz_ *)
%Y A346538 Main diagonal gives A346539.
%Y A346538 Column (or row) k=0 gives A347814.
%Y A346538 Cf. A008288, A001850, A001263, A006318, A001006, A114486.
%K A346538 nonn,tabl
%O A346538 0,4
%A A346538 _José María Grau Ribas_, Jul 23 2021