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.

1, 3, 25, 241, 2545, 28203, 322681, 3776275, 44947503, 542097295, 6607714859, 81247609095, 1006335719467, 12542292874825, 157159924565801, 1978517963096763, 25010881408459855, 317327992746937599, 4039340709637022007, 51569571332132589961, 660140626022179390983

Offset: 0

Author

José María Grau Ribas, Sep 10 2021

Keywords

• nonn

Comments

All terms are odd.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..886
• Alois P. Heinz, Animation of a(3) = 241 paths

Crossrefs

Main diagonal of A346538.

Column k=2 of A347811.

Cf. A008288, A001850, A225042.

Programs

Maple

b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
     `if`(i^2+j^2 b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 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[{n,n}],{n, 0,22}]
(* Second program: *)
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}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)

Formula

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