In[]:=
NestGraph[n{2n,n+1},0,16]
Out[]=
In[]:=
ResourceFunction["GraphNeighborhoodVolumes"][NestGraph[n{2n,n+1},0,12]]
In[]:=
First[Values[%53]]
Out[]=
{1,2,3,5,8,13,21,34,55,89,144,233,377}
In[]:=
ResourceFunction["RaggedMeanAround"][Values[%53]]
Out[]=
{1,2.991
±
0.006
,6.57
±
0.06
,11.97
±
0.19
,19.8
±
0.5
,29.6
±
1.0
,42.2
±
1.8
,57.8
±
3.0
,76.
±
5.
,95.
±
7.
,113.
±
9.
,122.
±
9.
,129.
±
9.
,133.
±
8.
,146.
±
9.
,154.
±
9.
,160.
±
9.
,171.
±
9.
,175.
±
9.
,179.
±
9.
,189.
±
9.
,199.
±
10.
,203.
±
9.
,206.
±
9.
,215.
±
9.
,218.
±
9.
,221.
±
8.
,230.
±
8.
,239.
±
8.
,242.
±
8.
,244.
±
7.
,252.
±
7.
,254.
±
7.
,256.
±
6.
,265.
±
7.
,273.
±
7.
,281.
±
7.
,284.
±
7.
,286.
±
7.
,293.
±
7.
,295.
±
6.
,297.
±
6.
,305.
±
6.
,313.
±
6.
,315.
±
6.
,318.
±
5.
,325.
±
4.
,329.5
±
0.5
}
In[]:=
ListLinePlot[%]
Out[]=
In[]:=
ResourceFunction["GraphNeighborhoodVolumes"][NestGraph[n{2n,n+1},0,14]];
In[]:=
ResourceFunction["RaggedMeanAround"][Values[%]]
Out[]=
{1,2.9967
±
0.0023
,6.60
±
0.04
,12.05
±
0.11
,19.96
±
0.29
,30.1
±
0.6
,43.0
±
1.1
,59.4
±
2.0
,79.6
±
3.2
,101.
±
5.
,125.
±
7.
,151.
±
9.
,177.
±
12.
,191.
±
12.
,204.
±
13.
,214.
±
12.
,236.
±
14.
,246.
±
14.
,256.
±
14.
,273.
±
14.
,283.
±
14.
,292.
±
14.
,309.
±
15.
,326.
±
16.
,335.
±
16.
,344.
±
16.
,360.
±
16.
,369.
±
16.
,378.
±
16.
,395.
±
16.
,412.
±
17.
,421.
±
17.
,429.
±
16.
,443.
±
17.
,448.
±
16.
,454.
±
16.
,467.
±
16.
,481.
±
17.
,494.
±
17.
,499.
±
17.
,504.
±
17.
,516.
±
17.
,521.
±
17.
,525.
±
16.
,537.
±
17.
,550.
±
17.
,554.
±
16.
,558.
±
16.
,570.
±
16.
,574.
±
16.
,578.
±
15.
,590.
±
15.
,602.
±
16.
,614.
±
16.
,618.
±
15.
,622.
±
15.
,633.
±
15.
,636.
±
14.
,640.
±
14.
,651.
±
14.
,662.
±
14.
,666.
±
13.
,669.
±
13.
,680.
±
13.
,683.
±
12.
,686.
±
12.
,697.
±
12.
,708.
±
13.
,719.
±
13.
,730.
±
14.
,734.
±
13.
,737.
±
13.
,747.
±
13.
,750.
±
13.
,753.
±
13.
,763.
±
13.
,773.
±
13.
,776.
±
13.
,779.
±
12.
,789.
±
12.
,791.
±
12.
,794.
±
11.
,804.
±
12.
,815.
±
12.
,825.
±
12.
,828.
±
12.
,831.
±
11.
,841.
±
11.
,843.
±
11.
,846.
±
11.
,856.
±
10.
,867.
±
9.
,870.
±
9.
,874.
±
9.
,885.
±
7.
,891.5
±
0.5
}
In[]:=
ListLinePlot[%]
Out[]=
In[]:=
ResourceFunction["LogDifferences"][ResourceFunction["RaggedMeanAround"][Values[%61]]]
Out[]=
{1.5834
±
0.0011
,1.948
±
0.014
,2.09
±
0.04
,2.26
±
0.08
,2.24
±
0.14
,2.33
±
0.22
,2.42
±
0.32
,2.5
±
0.4
,2.3
±
0.6
,2.2
±
0.8
,2.2
±
1.0
,2.0
±
1.1
,1.0
±
1.2
,0.9
±
1.3
,0.8
±
1.3
,1.6
±
1.3
,0.7
±
1.4
,0.7
±
1.4
,1.3
±
1.5
,0.7
±
1.5
,0.7
±
1.5
,1.3
±
1.5
,1.3
±
1.6
,0.7
±
1.6
,0.7
±
1.7
,1.3
±
1.7
,0.7
±
1.7
,0.7
±
1.7
,1.3
±
1.7
,1.3
±
1.8
,0.7
±
1.8
,0.7
±
1.8
,1.0
±
1.8
,0.4
±
1.8
,0.4
±
1.8
,1.1
±
1.8
,1.1
±
1.9
,1.0
±
1.9
,0.4
±
1.9
,0.4
±
1.9
,1.0
±
1.9
,0.4
±
2.0
,0.4
±
1.9
,1.0
±
1.9
,1.0
±
2.0
,0.4
±
2.0
,0.4
±
1.9
,1.0
±
1.9
,0.4
±
1.9
,0.4
±
1.9
,1.1
±
1.9
,1.0
±
1.9
,1.0
±
1.9
,0.4
±
1.9
,0.3
±
1.9
,1.0
±
1.9
,0.3
±
1.9
,0.3
±
1.8
,1.0
±
1.8
,1.0
±
1.8
,0.3
±
1.8
,0.3
±
1.7
,1.0
±
1.7
,0.3
±
1.7
,0.3
±
1.6
,1.1
±
1.7
,1.1
±
1.7
,1.1
±
1.8
,1.0
±
1.8
,0.3
±
1.8
,0.3
±
1.8
,1.0
±
1.8
,0.3
±
1.8
,0.3
±
1.8
,1.0
±
1.8
,1.0
±
1.8
,0.3
±
1.8
,0.3
±
1.8
,1.0
±
1.7
,0.3
±
1.7
,0.3
±
1.7
,1.1
±
1.7
,1.1
±
1.7
,1.0
±
1.7
,0.3
±
1.7
,0.3
±
1.7
,1.0
±
1.7
,0.3
±
1.6
,0.3
±
1.6
,1.1
±
1.6
,1.1
±
1.5
,0.4
±
1.4
,0.4
±
1.4
,1.1
±
1.2
,0.7
±
0.7
}
In[]:=
ListLinePlot[%]
Out[]=
In[]:=
ResourceFunction["GraphNeighborhoodVolumes"][NestGraph[n{2n,2n+1},0,12]];
In[]:=
ListLinePlot[ResourceFunction["LogDifferences"][ResourceFunction["RaggedMeanAround"][Values[%]]]]
Out[]=

QuotientRemainder Plots

Modulo

Collatz without choice

Reversible 3n+1

https://www.wolframscience.com/nks/notes-4-2--a-reversible-3n1-problem-system/

Building integers from operations

https://www.wolframscience.com/nks/notes-4-5--operator-representations/

Mod cases

Universal Arithmetic System

https://www.wolframscience.com/nks/p673--emulating-cellular-automata-with-other-systems/

Algebraic Number

UFDs?

< Like arithmetic series , or geometric series >