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A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.

%I A346539 #32 Nov 03 2021 08:42:19
%S A346539 1,3,25,241,2545,28203,322681,3776275,44947503,542097295,6607714859,
%T A346539 81247609095,1006335719467,12542292874825,157159924565801,
%U A346539 1978517963096763,25010881408459855,317327992746937599,4039340709637022007,51569571332132589961,660140626022179390983
%N A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.
%C A346539 All terms are odd.
%H A346539 Alois P. Heinz, <a href="/A346539/b346539.txt">Table of n, a(n) for n = 0..886</a>
%H A346539 Alois P. Heinz, <a href="/A346539/a346539_1.gif">Animation of a(3) = 241 paths</a>
%F A346539 a(n) ~ c * d^n / n^(3/2), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.56165398271839628518... and c = 2.3842296614800994817864695565477260682981556338086519... . - _Vaclav Kotesovec_, Sep 13 2021
%p A346539 b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
%p A346539      `if`(i^2+j^2<n^2+k^2, b(sort([i, j])[]), 0), j=k-1..k+1), i=n-1..n+1))
%p A346539     end:
%p A346539 a:= n-> b(n$2):
%p A346539 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 12 2021
%t A346539 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 A346539 $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[{n,n}],{n, 0,22}]
%t A346539 (* Second program: *)
%t A346539 b[n_, k_] := b[n, k] = If[{n, k} == {0, 0}, 1, Sum[Sum[If[i^2 + j^2 < n^2 + k^2, b@@Sort[{i, j}], 0], {j, k-1, k+1}], {i, n-1, n+1}]];
%t A346539 a[n_] := b[n, n];
%t A346539 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Nov 03 2021, after _Alois P. Heinz_ *)
%Y A346539 Main diagonal of A346538.
%Y A346539 Column k=2 of A347811.
%Y A346539 Cf. A008288, A001850, A225042.
%K A346539 nonn
%O A346539 0,2
%A A346539 _José María Grau Ribas_, Sep 10 2021