Description
Check an exact Pascal recurrence residual.
Exact combinatorial recurrences distinguish individual summands, adjacent ratios and complete totals. A local arithmetic identity can be correct without proving that its supplied inputs are the intended counts or that a whole enumeration has been completed.
Check an exact Pascal recurrence residual.
When to use this tool
Use Binomial Pascal Single Recurrence Residual when checking the stated recurrence, coefficient-ratio or inclusion-exclusion term.
How to use the tool and how it works
- Enter current, left, right. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read residual, valid. Inspect valid before interpreting the associated values.
Check an exact Pascal recurrence residual. Parse current/left/right decimal integers; residual = current-left-right as exact decimal text, valid means zero.1
Pascal recurrence and adjacent-coefficient ratios are exact combinatorial identities; checking a supplied local identity does not verify a complete row.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| current | Supplied current value under the stated calculation or update convention. | Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. |
| left | Left | Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. |
| right | Right | Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| residual | Signed difference in the direction stated above, with matching units. | Required. Text; maximum UTF-16 length 8192. |
| valid | Exact current-left-right equals zero; no assertion that inputs are actual nonnegative binomial coefficients. | Required. Boolean. |
{
"input": {
"current": "10",
"left": "4",
"right": "6"
},
"output": {
"residual": "0",
"valid": true
}
}Limitations and assumptions
- Negative values are allowed by the arithmetic contract; no binomial indexes or actual Pascal coefficients are verified.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Alternative or Complementary methods
Use the complete recurrence with justified initial conditions and exact integer arithmetic when a full counting result is required.
References
Similar or alternative tools
- Binomial Lattice Step Forward Ratio
Return the forward ratio of adjacent binomial coefficients.
- Binomial Coefficient Log Ratio Step
Return adjacent binomial ratio and its natural logarithm.