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 >