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