A308114 Total number of nodes summed over all lattice paths from (0,0) to (n,n) that do not go above the diagonal x=y and consist of steps (h,v) with min(h,v) > 0 and gcd(h,v) = 1.

1, 2, 3, 7, 26, 92, 314, 1055, 3589, 12410, 43356, 152336, 537721, 1906063, 6781737, 24206994, 86644157, 310871212, 1117741815, 4026430097, 14528792287, 52504325068, 189999731589, 688411569408, 2497081766875, 9067028323162, 32953990726244, 119875216666167

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..550

Wikipedia, Counting lattice paths

Formula

a(n) ~ c * d^n / sqrt(n), where d = 3.7137893481485186502229788321701955452444... and c = 0.243302622746026118665161170169985306... - Vaclav Kotesovec, May 24 2019

Maple

b:= proc(x, y) option remember; `if`(y=0, [1$2], (p-> p+
      [0, p[1]])(add(add(`if`(x+v>y+h or igcd(h, v)>1, 0,
       b(x-h, y-v)), v=1..y), h=1..x)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..30);

Mathematica

f[p_List] :=  p + {0, p[[1]]}; f[0] = 0;
b[{x_, y_}] := b[{x, y}] = If[y == 0, {1, 1},
     f[Sum[Sum[If[x + v > y + h || GCD[h, v] > 1, {0, 0},
     b[{x - h, y - v}]], {v, 1, y}], {h, 1, x}]]];
a[n_] := b[{n, n}][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)

Crossrefs

Cf. A308112, A308113.

Sequence in context: A301519 A274692 A342155 * A092983 A089708 A107881

Adjacent sequences: A308111 A308112 A308113 * A308115 A308116 A308117

Keyword

nonn

Author

Alois P. Heinz, May 13 2019

Status

approved