Sequence Calculator

What does this sequence calculator compute?

Closed forms for arithmetic and geometric sequences, iterative Fibonacci F₀=0 F₁=1, and uₙ = p uₙ₋₁ + q uₙ₋₂. Discovery pages share this capability.

The published formula or method is a_n = a+(n−1)d or a r^{n−1}; F_n recurrence. The following references support the stated mathematics or method:

• Margaret Stevens — The Maths Workbook (University of Oxford): 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.
• MIT Mathematics for Computer Science — Fibonacci Recurrences: 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.

Which inputs and limits should you check?

Check the sequence type, starting-index convention, and finite-term count. Preserve complete terms and exact integers where the result provides them.

The worked example uses a=2 d=3 n=5 with 2,5,8,11,14. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.sequences v1.0.3 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Exact integer scope and verification

This entry belongs to math.sequences and its four modes: arithmetic, geometric, Fibonacci and second-order linear recurrence. AP/GP use indices 1–1000; Fibonacci/recurrence allow 0–1000. F₀=0, F₁=1; the recurrence is uₙ=p·uₙ₋₁+q·uₙ₋₂ with supplied a₀,a₁. The Fibonacci sum is F₀+…+Fₙ; recurrence mode returns the term without claiming a partial sum. GP handles r=0,1,−1 explicitly; for n=1 the first term is returned even when r=0.

Coefficients are signed integer digit strings of at most 48 digits, and every raw token is at most 512 characters before trimming. Index tokens contain at most 4 digits. Leading zeros count toward the limit. Finite safe JSON integers are also accepted; fractional, unsafe numeric, decimal/scientific string, hex, boolean and null inputs are rejected. Explicit empty input is an error. Aliases must agree; unknown and inactive fields are rejected. Preset entries lock their mode and reject a conflicting mode.

BigInt calculations are exact relative to accepted integers; there is no floating-point rounding tolerance. Results within ±9007199254740991 are JSON numbers; larger integers are exact decimal strings. All term/value/exact, applicable sum/previous term, conventions and diagnostics remain in the complete JSON/CSV export. Large results are fully available in the result and work. Relative sensitivity can be large when p/q or positive/negative terms nearly cancel; exact accepted inputs do not remove uncertainty in measured inputs. Noninteger, real-valued and infinite series are outside this calculator's domain.

The stem figure contains at most16 ending indices. It is omitted with an explicit notice when any plotted integer exceeds safe coordinates or distinct terms cannot be resolved at the chosen scale. Text and exports retain the full integers; absence of a graph does not mean zero or no solution.

The independent reference uses Python arbitrary integers: AP/GP enumeration and matrix powers for Fibonacci/recurrence, unlike the production closed forms/iterative recurrences. The current fixed suite has 37 independent complete packs, strict keys/types/values/metadata and comparator negative controls. This scope is not exhaustive proof over all inputs or third-party certification. Previous1.0.1 verification pins remain immutable.

Typing calculates locally. Shared URLs contain inputs and send them to the host; REST/MCP send supplied inputs. Browser/server logs may retain requests. No zero-log or whole-site network privacy guarantee is made.

Model definitions: NIST Fibonacci recurrence and Western University sequences and finite sums. These sources explain mathematics, not implementation certification.