OFFSET
0,2
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
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
FORMULA
a(n) >= 2*a(n-1) for n > 0.
EXAMPLE
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:
__ __.__ __|__|
|__| |__.__| |__|
PROG
(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<<logint(f, 2)); f-=t; self()(w, f, b, r); self()(w-1, f, bitor(b, t), r); self()(w-1, f, bitor(b, 2*t), r); self()(w-2, f, bitor(b, 3*t), r) )));
mapput(M, [n, 1], 1);
for(n=1, n, my(L=Mat(M)); M=Map();
for(i=1, matsize(L)[1], my([hk, r]=L[i, ]); recurse(hk[1], hk[2], 0, r)));
mapget(M, [0, 1]);
} \\ Andrew Howroyd, May 24 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Roman Hros, May 23 2021
EXTENSIONS
Terms a(25) and beyond from Andrew Howroyd, May 24 2021
STATUS
approved