Explore how simple rules generate structured numerical patterns and intersections across different bases.

Sequence Intersection Explorer

Explore how simple rules generate structured numerical patterns and intersections across different bases

Base B1

Base B2

Max index N

Intersection filter Default view: nontrivial

Chart mode Largest labels only

The chart highlights the selected intersections only. A true crossing happens only when the same value occurs at the same index.

Results

Total intersections within N: –

Selected intersections: –

Type: –

Largest selected value: –

Insight

P = –

–

Intersection parameter P = (B2 − B1)(B1 + B2 − 7).

Intersection list (selected filter, within N)

Value	(n1, n2)	Phase type	Same index?

To understand the basics and how the system works, you can read or download the entire development and study from here Zenodo.

How to Use the Tool

This tool allows you to explore intersections between sequences generated by the Progressive Cumulative Counting (PCC) model.

Each sequence is defined by a base $B$ B. After an initial linear phase, the sequence enters a cumulative growth phase where values increase according to a structured rule related to triangular numbers.

How to use the explorer

Select two bases (B1 and B2)
Use the sliders to choose the bases of the two sequences you want to compare.
Set the maximum index (N)
This determines how far the sequences are computed and analyzed.
Click “Compute”
The tool will generate the two sequences and search for values that appear in both.

Interpreting the results

The graph shows the two sequences as curves.
Green points indicate values that appear in both sequences.
When the same value occurs at the same index, the curves intersect directly.
When the value appears at different indices, the points share the same height but occur at different horizontal positions.

Filters

You can filter intersections by type:

Nontrivial – excludes simple overlaps from the initial phase
Cumulative – intersections occurring after the cumulative growth phase
Mixed – intersections between different phases
Bootstrap – intersections in the initial phase
All – shows every detected intersection

Additional information

The panel also shows the parameter P = (B₂ − B₁)(B₁ + B₂ − 7), which describes the structural relationship between the two sequences and helps interpret whether intersections are expected to be finite or structurally special.

This interactive explorer helps reveal how simple generative rules can produce structured numerical patterns and shared values across different bases.

The tool above was built for demonstration purposes, in addition to the graph, read the part below “intersection list” so that everything is clearer.