In[]:=

NestGraph[n|->{2n,n+1},1,5,VertexLabels->Automatic]

Out[]=

In[]:=

NestGraph[n|->{2n,n+1},1,10]

Out[]=

In[]:=

NestGraph[n|->{2n,n+1},{1,2},2]

Out[]=

In[]:=

NestGraph[n|->{2n,n+1},{2,3},3]

Out[]=

In[]:=

NestGraph[n|->{2n,n+1},1,10,VertexLabelsAutomatic]

Out[]=

Pentagon:

{n,n+1,2n,2n+1,2n+2}

Odd nodes: 3 edges

Even nodes: 4 edges

Even nodes: 4 edges

Seems to be a spiral

In[]:=

UndirectedGraph[NestGraph[n|->{2n,n+1},1,10]]

Out[]=

## Other Cases

Other Cases

Since this is a linear function of n, it doesn’t matter whether we start from 1 or x.....

This compares the two branches of the cycle:

Iterating down two branches with

do these ever match?

This is the same as 2n , n + 1

In {a n, n+1} appears to be making (a+3)-gons.......

One branch gives powers of a; the other gives successive numbers....

In the ‘gons, claim is that there are two “a events”, and the others are all +1 events

Directions are all from smallest number as source, to largest number as sink [ inevitable because this rule only makes numbers larger ]

## Other Cases

Other Cases

Shows which event is which:

The two “blue branches” correspond to:

These take a steps to match up “on the other side”

### Agreement of iterations

Agreement of iterations

Basic question: when does a number first recur?

Length 3 + length 2 pentagon

Nested addition ; nested multiplication : both just powers

Non-interleaved case....