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  "api": "https://www.calculatorx.com/api/v1/calc/sequence",
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  "schema_output": "https://www.calculatorx.com/schemas/math.sequences.output.json",
  "assumptions": [
    "The Result Card includes an engine-linked stem figure of a_k (marker is published term; at most 16 indices, window ending at n). Same closed form / recurrence as REST/MCP. Terms array is not a REST field. Huge BigInt terms omit the figure.",
    "Integers",
    "n ≤ 1000",
    "Not Pascal / partition / n!"
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    "Digit strings only;unsafe JSON numbers rejected. Aliases must agree;unknown andinactive inputs rejected. Results use JSON numbers only within safe integers, otherwise exact decimal strings.",
    "37 independent Python integer-enumeration/matrix-power full packs;exact complete structure/type/metadata comparison,notthird-party certification.",
    "Large orindistinguishable exact terms omit graph explicitly;fullintegers remain inresult/work/nativeexports. Continuousreal-valued ornonintegersequences unsupported."
  ],
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    "unknown mode (including pascal) → INVALID_MODE",
    "n=0 on AP/GP → VALUE_MUST_BE_POSITIVE",
    "n>1000 → VALUE_ABOVE_MAX"
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  "sources": [
    {
      "title": "ISO 80000-2:2019 Quantities and units — Mathematics",
      "url": "https://www.iso.org/standard/64973.html",
      "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.",
      "kind": "standard",
      "publisher": "International Organization for Standardization",
      "reference_scope": "all"
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    {
      "title": "Knuth, The Art of Computer Programming, Fibonacci F₀=0 F₁=1 and linear recurrences",
      "supports": "F_n = F_{n-1}+F_{n-2} with F_0=0; u_n = p u_{n-1} + q u_{n-2}",
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      "oracle_class": "O1"
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    {
      "title": "Margaret Stevens — The Maths Workbook (University of Oxford)",
      "section": "Chapter 3, §2: Arithmetic and Geometric Sequences and Series",
      "url": "https://www.some.ox.ac.uk/wp-content/uploads/2024/08/Maths-Workbook.pdf",
      "supports": "Arithmetic terms and sums from first term and common difference, geometric terms and finite sums from first term and common ratio, including derivations. At ratio 1 the finite geometric sum is n times the first term; it is the removable special case of the quoted expression. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "University of Oxford, Department of Economics; hosted by Somerville College"
    },
    {
      "title": "MIT Mathematics for Computer Science — Fibonacci Recurrences",
      "section": "§15.4 Solving Linear Recurrences",
      "url": "https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_textbook.pdf",
      "supports": "The Fibonacci recurrence with F₀=0 and F₁=1, its generating function and Binet expression. The recurrence identifies the mathematical sequence; it does not independently test a fast-doubling implementation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "Massachusetts Institute of Technology"
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