Description
Return shiftEntry one only for row>0 and column=row-1, and coefficientRow for row zero.
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 shiftEntry one only for row>0 and column=row-1, and coefficientRow for row zero.
When to use this tool
Use Recurrence Companion State Index Map 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 order, row, column. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read shiftEntry, coefficientRow. Inspect coefficientRow before interpreting the associated values.
Return shiftEntry one only for row>0 and column=row-1, and coefficientRow for row zero. Require order one through 256 and bounded zero-based row and column.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| order | Ordering sign or index under the specific convention stated above. | Required. Safe integer; minimum 1; maximum 256. |
| row | Row | Required. Safe integer; minimum 0. |
| column | Column | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| shiftEntry | Shift Entry | Required. Safe integer. |
| coefficientRow | True only for row zero in a newest-first companion layout. | Required. Boolean. |
{
"input": {
"order": 4,
"row": 2,
"column": 1
},
"output": {
"shiftEntry": 1,
"coefficientRow": false
}
}Limitations and assumptions
- This uses a newest-first companion-state layout. The coefficient row is identified but its numeric recurrence coefficient is not supplied or returned; no complete matrix is constructed.
- 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
- Recurrence Delay State Shift Index
Return nextIndex=index+1 for a bounded state index except the final position, which returns minus one and dropped true.
- Lucas Recurrence One State Update
Parse current, previous, p, and q as bounded exact decimal integers and return next=p*current-q*previous as decimal text.