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

0,2

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

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

Maple

b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
     `if`(i^2+j^2<n^2+k^2, b(sort([i, j])[]), 0), j=k-1..k+1), i=n-1..n+1))
    end:
a:= n-> 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 *)

Crossrefs

Main diagonal of A346538.

Column k=2 of A347811.

Cf. A008288, A001850, A225042.

Sequence in context: A332468 A134272 A335117 * A350224 A171640 A099913

Adjacent sequences: A346536 A346537 A346538 * A346540 A346541 A346542

Keyword

nonn

Author

José María Grau Ribas, Sep 10 2021

Status

approved