Description
Calculate a polygonal-number discriminant and numerical inverse index, with a floating-point integrality check.
A sequence rule specifies how terms relate. Checking one transition, advancing a recurrence, and bounding an infinite tail answer different questions. A calculation from supplied coefficients assumes that rule; it does not infer it from observations.
Calculate a polygonal-number discriminant and numerical inverse index, with a floating-point integrality check.
When to use this tool
Use it for figurate sequence inverse index discriminant when the required inputs and conventions are already available.
How to use the tool and how it works
- Enter sides, value. Use the stated units and boundary conventions.
- Run the calculation. Correct any input error before interpreting a result.
- Read discriminant, index, integralIndex. Inspect any known, defined, inclusion, or boundary flags before interpreting numeric values.
Invert the supplied polygonal-number formula using discriminant (sides-4)^2 + 8*(sides-2)*value. Return its exact decimal text and the positive-branch numerical index, special-casing value zero to index zero, then test whether that computed index is an integer.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| sides | Sides | Required. Safe integer; minimum 3; maximum 1000. |
| value | Value | Required. Safe integer; minimum 0; maximum 1000000. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| discriminant | Discriminant | Required. Text; maximum UTF-16 length 8192. |
| index | Numerical positive-branch inverse polygonal index, with value zero handled separately. | Required. Finite number. |
| integralIndex | Integral Index | Required. Boolean. |
{
"input": {
"sides": 3,
"value": 15
},
"output": {
"discriminant": "121",
"index": 5,
"integralIndex": true
}
}Limitations and assumptions
- Sides is 3–1000 and value is 0–1,000,000. integralIndex checks the computed floating-point value rather than providing a symbolic square-root proof or validating arbitrary figurate sequences.
- The field tables describe structural bounds; the mechanics also state cross-field requirements. Malformed input shapes, disallowed keys, and invalid domains cause input errors. Results must remain finite and within the declared output contract. A successful false flag or zero placeholder is distinct from an error.
- Numeric arithmetic uses binary floating point. Exact comparisons have no hidden tolerance or decimal rounding; negative zero is exported as zero. Decimal-text integer operations retain the exactness specifically stated above.
Alternative or Complementary methods
For figurate sequence inverse index discriminant, check the source records and the supplied rule before comparing the result with another method. Use exact integer or symbolic arithmetic when a numerical equality is insufficient.
References
Similar or alternative tools
- Factorial Sequence Ratio At Index
Return the adjacent factorial ratio or its reciprocal without constructing factorials.
- Affine Recurrence Two Step Constant
Return the coefficients obtained by composing a supplied affine recurrence twice.