A308087 Number of lattice paths from (0,0) to (n,n) using Euclid's orchard as a step-set.

1, 1, 1, 3, 13, 45, 153, 515, 1767, 6167, 21697, 76661, 271973, 968561, 3460677, 12399661, 44534647, 160285049, 577949447, 2087375443, 7550053527, 27344761057, 99155777619, 359943568005, 1307923066305, 4756914915657, 17315390737219, 63077564876055

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..575

J. East and N. C. Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.

N. C. Ham, Implementation of algorithms 1-3 from J. East and N. C. Ham reference above.

Wikipedia, Euclid's orchard

Formula

a(n) mod 2 = 1. - Alois P. Heinz, May 13 2019

a(n) ~ c * d^n / sqrt(n), where d = 3.7137893481485186502229788321701955452444... and c = 0.133597878112414800677299372849715598093... - Vaclav Kotesovec, May 24 2019

Maple

b:= proc(x, y) option remember; `if`(y=0, 1, add(add(`if`(1=
      igcd(h, v), b(sort([x-h, y-v])[]), 0), v=1..y), h=1..x))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 12 2019

Mathematica

b[x_, y_] := b[x, y] = If[y == 0, 1, Sum[Sum[If[1 == GCD[h, v], b @@ Sort[{x - h, y - v}], 0], {v, 1, y}], {h, 1, x}]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)

Crossrefs

Cf. A001764, A005043, A005165, A035343, A067955, A097609, A308112, A308113.

Sequence in context: A187915 A115128 A140420 * A232231 A121136 A350175

Adjacent sequences: A308084 A308085 A308086 * A308088 A308089 A308090

Keyword

nonn,walk

Author

Nicholas Ham, May 11 2019

Extensions

a(16)-a(27) from Alois P. Heinz, May 12 2019

Status

approved