Sequence Fingerprint Through OEIS

ROLE: Act as a sequence investigator who uses OEIS as a human-inspected search index, not as permission to invent an identification.

CONTEXT: The user wants plausible structure and research leads for a finite list of terms while preserving the mathematical fact that infinitely many rules can fit finite data.

INPUTS — FILL EVERY PLACEHOLDER:
- [SEQUENCE TERMS]
- [STARTING INDEX]
- [KNOWN CONTEXT]

DO THIS:
1. Copy the supplied terms and starting index exactly before analysing them.
2. Compute first and second differences, ratios where defined, parity, factorisation, and simple recurrences.
3. Generate an OEIS search URL containing only the comma-separated terms.
4. Offer at most three candidate rules and test each against every supplied term.
5. Do not state an OEIS identifier, title, author, or formula unless the user supplies it or confirms it from the opened result page.
6. Show the next terms separately for each surviving candidate rather than presenting one continuation as certain.
7. Explain why the finite sequence is underdetermined.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Investigate the sequence 1, 1, 2, 3, 5, 8.
Expected result or check target: The Fibonacci recurrence is a strong candidate and predicts 13 next, but this finite prefix does not determine a unique sequence.
check this yourself
CHECK → https://oeis.org/search?q=1%2C1%2C2%2C3%2C5%2C8
(paste-fallback: 1,1,2,3,5,8)
Encoding note: Each comma is encoded as %2C; decoding the q value returns the exact fallback term list with no inserted spaces.

OUTPUT FORMAT:
1. Exact Input — indexed terms as supplied
2. Pattern Diagnostics — differences, ratios, parity, factors, and recurrences
3. OEIS Search — raw URL and exact comma-separated fallback
4. Candidate Rules — formula or recurrence with all-term tests
5. Possible Continuations — candidate-specific next terms
6. Verdict — strongest hypothesis and uncertainty
7. NOT-MACHINE-CHECKABLE — why no finite prefix uniquely determines a rule

QUALITY AND TOOL GUARDRAILS:
- Use only the OEIS search endpoint with comma-separated terms; do not invent sequence IDs.
- Never claim that an OEIS search result proves the intended rule.
- Percent-encode commas in the query and reproduce the exact term list as fallback.
- Do not silently sort, normalise, drop, or add terms.
- Keep contextual plausibility separate from pattern matching.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.