Description
Return gap=(sides-2)*index+1 for integer sides three through one million and index zero through one million.
A sequence rule specifies an index origin, state order, and relation among terms. Advancing one step, mapping an index, and checking a local residual answer different questions. Supplied coefficients or periods define a model; they do not establish that observations follow it.
Return gap=(sides-2)*index+1 for integer sides three through one million and index zero through one million.
When to use this tool
Use Polygonal Number Adjacent Gap when you need this particular index map, recurrence step, count, or local algebraic term under supplied sequence conventions.
How to use the tool and how it works
- Enter sides, index. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read gap. Use the output convention below.
Return gap=(sides-2)*index+1 for integer sides three through one million and index zero through one million.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| sides | Sides | Required. Safe integer; minimum 3; maximum 1000000. |
| index | Zero-based index unless the mechanics explicitly state otherwise. | Required. Safe integer; minimum 0; maximum 1000000. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| gap | Gap | Required. Safe integer. |
{
"input": {
"sides": 5,
"index": 3
},
"output": {
"gap": 10
}
}Limitations and assumptions
- This is the adjacent polygonal-number gap under the supplied zero-based index convention. The tool does not invert a polygonal number, verify membership, or enumerate the sequence.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Choose exact integer arithmetic when the index or coefficient range requires it. A complete sequence analysis needs actual terms and a consistent index and state-order convention beyond one local step.
References
Similar or alternative tools
- Interleaved Sequence Odd Even Length
Return evenPositions=ceil(length/2) and oddPositions=floor(length/2) for nonnegative integer length.
- Linear Recurrence Coefficient Impulse Term
Return contribution=coefficients[index]*impulse for a valid zero-based index.