A344571 Number of subgraphs of the directed square lattice with n edges and all vertices reachable from the origin.

1, 2, 5, 14, 42, 130, 412, 1326, 4318, 14188, 46950, 156258, 522523, 1754254, 5909419, 19964450, 67618388, 229526054, 780633253, 2659600616, 9075301990, 31010850632, 106100239080, 363428599306, 1246172974048, 4277163883744, 14693260749888, 50516757992258

Offset: 0

Author

Roman Hros, May 23 2021

Keywords

• nonn

Comments

Equivalently, the number of fixed polysticks (see A096267) that can be constructed starting from a fixed vertex by only adding edges on top of an existing vertex or to the right of an existing vertex. If the polystick is rotated counterclockwise by 45 degrees, then the polystick is supported from the starting vertex. - Andrew Howroyd, May 24 2021

Examples

			In the following examples, the origin is in the bottom left corner and graph edges are directed upwards and to the right.
The a(1) = 2 graphs are:
  |   __
.
The a(2) = 5 graphs are:
  |   __
  |  |     __.__    __|   |__
.
The a(3) = 14 graphs are:
  |    __
  |   |     |__    __|    __.__    |      __
  |   |     |     |      |         |__   |__
.
                               __    |
  __.__.__   __.__|  __|__  __|    __|   |____  |_|
.
Other examples with 4, 6, and 7 edges respectively include:
     __      __.__      __|__|
    |__|    |__.__|    |__|
		

Links

• Roman Hros, Generating Subgraphs of Finite Grids, IIT.SRC 2018: 14th Student Research Conference in Informatics and Information Technologies.
• Eric Weisstein's World of Mathematics, Polystick
• Wikipedia, Polystick

Crossrefs

Cf. A096267, A353028.

Programs

PARI

a(n)={
  local(M=Map());
  my(acc(hk, r)=my(z); mapput(M, hk, if(mapisdefined(M,hk,&z),z+r,r)));
  my(recurse(w,f,b,r) =
    if(w<=0, if(w==0, acc([w,1],r)), if(f==0, if(b, acc([w,b>>valuation(b,2)],r)),
    my(t=1<Andrew Howroyd, May 24 2021

Formula

a(n) >= 2*a(n-1) for n > 0.

Extensions

Terms a(25) and beyond from Andrew Howroyd, May 24 2021