HomeCalculatorsMathNumber Theory & Discrete MathSequence Calculator
Math calculator

Sequence Calculator

Integer arithmetic and geometric sequences, Fibonacci (F₀=0), and linear recurrences. Exact BigInt. Not Pascal’s triangle. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 5/5 declared partitions (arithmetic, geometric, fibonacci, linear-recurrence, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
a_n / F_n / u_n for integer sequences, plus S_n for AP/GP/Fibonacci, with an engine-linked stem figure of a_k through n.
Scope
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.
Verification
Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.signed-bigint-publication
Verification revision
2026-10-07.signed-bigint-publication · 3/3 property · digest 0c1fda7d6011
Legacy regression
12/12 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
8/8 golden · 3/3 CVP boundary · 6/6 invalid · 3/3 property · 5/5 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • Knuth, The Art of Computer Programming, Fibonacci F₀=0 F₁=1 and linear recurrences
Sources
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 8/8 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Arithmetica_n = a + (n−1)d
Geometrica_n = a r^{n−1}
FibonacciF_n = F_{n−1} + F_{n−2}
iIntegers only. Exact BigInt. n ≤ 1000. Not Pascal’s triangle, not integer partition, not n!. Engine id is sequence — never the add/sub arithmetic engine.

How to use

1

Choose a mode

Arithmetic, geometric, Fibonacci, or linear recurrence. Locked discovery URLs exist for the first three.

2

Enter n and the parameters

Fibonacci allows n=0. Arithmetic/geometric need n≥1.

3

Read a_n (and S_n)

Sums are exact integers. The figure shows a stem plot of a_k through n.

Example calculations

Common configurations with formula and result.

ϟ

AP

a=2 d=3 n=5

2,5,8,11,14
14
ϟ

GP

a=3 r=2 n=4

3,6,12,24
24
ϟ

F_10

F_0=0

F_10
55

Sequence calculator specification

Version 1.0.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
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. Not combinatorics.
What it calculates
a_n / F_n / u_n for integer sequences, plus S_n for AP/GP/Fibonacci, with an engine-linked stem figure of a_k through n.
Inputs
  • mode
  • n
  • a
  • d
  • r
  • p
  • q
  • a0
  • a1
Outputs
  • term
  • sum
  • exact
Formula
a_n = a+(n−1)d or a r^{n−1}; F_n recurrence
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!
Units
  • dimensionless
Boundary conditions
  • unknown mode (including pascal) → INVALID_MODE
  • n=0 on AP/GP → VALUE_MUST_BE_POSITIVE
  • n>1000 → VALUE_ABOVE_MAX
Example
mode=arithmetic a=2 d=3 n=5 → 14
Validation cases

2 published on this page · 12/12 tests · Production surface contract 6/6 · View evidence

  • mode=arithmetic a=2 d=3 n=5 → term=14 sum=40
  • mode=fibonacci n=10 → 55
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Sequence index n, arithmetic/geometric closed forms, and recurrence notation
  • 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}
Calculation version
1.0.1

Other calculators in this family: Arithmetic Sequence Calculator, Factorial Calculator, Fibonacci Calculator, GCF Calculator, Geometric Sequence Calculator, LCM Calculator, Modular Inverse Calculator, Modulo Calculator . Explore all Number Theory & Discrete Math.

→ Arithmetic Sequence Calculator Nth term and partial sum of an integer arithmetic sequence. Same engine as Sequence. Not geometric. Runs locally. › → Factorial Calculator Calculate n! for integers 0 ≤ n ≤ 5000. Fast local exact integers with shareable, machine-readable results. › → Fibonacci Calculator Fibonacci numbers F_n for n ≤ 1000. F_0=0, F_1=1. Same engine as Sequence. Exact BigInt. Runs locally. › → GCF Calculator Find the GCF, GCD, or HCF of two or more integers. Exact BigInt Euclidean fold with factor and prime cross-checks. Free, no sign-up. › → Geometric Sequence Calculator Nth term and partial sum of an integer geometric sequence. Same engine as Sequence. Not arithmetic. Runs locally. › → LCM Calculator Find the LCM of two or more integers with exact arithmetic and independent verification. Free, no sign-up. › → Modular Inverse Calculator Modular inverse of a modulo m when gcd(a,m)=1. Same engine as Euclidean remainder. Not a second GCF page. Runs locally. › → Modulo Calculator Integer remainder with a = q·m + r and 0 ≤ r < |m|. Euclidean convention, not JavaScript truncated %. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this Pascal’s triangle?

No. Binomial coefficients stay /calc/math/permutation-combination. Integer partition is out of scope.

Which Fibonacci indexing?

F_0=0, F_1=1. Linear recurrence with p=q=1, a0=0, a1=1 is the same sequence.

Where does this run?

Locally in the browser by default. REST and MCP call the same sequence engine.