A308112 Total number of nodes summed over all lattice paths from (0,0) to (n,n) that consist of steps (h,v) with min(h,v) > 0 and gcd(h,v) = 1.

1, 2, 3, 10, 47, 186, 703, 2640, 9979, 37980, 144713, 550666, 2093215, 7951524, 30186737, 114522342, 434172249, 1644889496, 6227677911, 23563691408, 89104756279, 336752825864, 1271998719875, 4802187032270, 18120902471019, 68347041380528, 257673014416775

Offset: 0

Author

Alois P. Heinz, May 13 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..550
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A059841, A308087, A308114.

Programs

Maple

b:= proc(x, y) option remember; `if`(y=0, [1$2],
      (p-> p +[0, p[1]])(add(add(`if`(igcd(h, v)=1,
       b(sort([x-h, y-v])[]), 0), 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[GCD[h, v] == 1,
     b[Sort[{x-h, y-v}]], {0, 0}], {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 *)

Formula

a(n) mod 2 = 1 - (n mod 2) = A059841(n).

a(n) ~ c * d^n * sqrt(n), where d = 3.7137893481485186502229788321701955452444... and c = 0.0685686817861124238901083560487601593693... - Vaclav Kotesovec, May 24 2019