Sequences & series
Arithmetic and geometric sequences, recurrences and their sums, with the general term made explicit. 18 tools, including Arithmetic sequence calculator, Catalan number generator, Collatz conjecture calculator and Cube numbers list. All free, with no account and nothing to install.
Arithmetic sequence calculatorFind the nth term and the sum of an arithmetic sequence.Catalan number generatorGenerate the nth Catalan number exactly, with big-integer precision. The Catalan numbers Cₙ = (2n)! / ((n+1)! n!) count an astonishing range of structures: balanced parentheses, binary trees, polygon triangulations, Dyck paths and hundreds more. Also lists C₀ through Cₙ.Collatz conjecture calculatorTrace the Collatz sequence from any positive integer: halve it if even, triple-plus-one if odd, and repeat until you reach 1. The famous unproven conjecture says every number eventually does. See the full trajectory, the number of steps and the peak value it soars to.Cube numbers listGenerate a list of perfect cubes — n³ for consecutive integers from any starting point. Set how many you want and where to start, and it returns the cubes, their sum and the underlying roots, computed with exact big-integer arithmetic.Fibonacci calculatorFind the nth number in the Fibonacci sequence.Fibonacci Number CheckerTest any number up to 60 digits with the exact 5n^2 +/- 4 identity, with its index, its neighbours in the sequence and the ratio compared with the golden ratio.Geometric sequence calculatorFind the nth term and the sum of a geometric sequence.Happy number calculatorFind out whether a number is 'happy'. Repeatedly replace it with the sum of the squares of its digits: if you reach 1 it is happy; if you fall into the 4-16-37-58-89-145-42-20 loop it is unhappy. See the verdict and the full chain of steps.Infinite Series Sum CalculatorTen classic families — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, Σ1/n! — with their closed form (π²/6, ln 2, π/4, e), the convergence condition and the partial sums.List of Fibonacci numbersGenerate the first n Fibonacci numbers exactly, with big-integer precision — each term the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8… The sequence hides the golden ratio and shows up throughout nature and mathematics. Returns the list and its sum.Partition function calculatorCompute p(n), the number of ways to write an integer as a sum of positive integers regardless of order — so 4 = 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 gives p(4) = 5. It uses Euler's pentagonal number recurrence with big integers, exact up to n = 500.Pascal's triangle generatorGenerate Pascal's triangle to any depth, with exact big-integer entries. Each number is the sum of the two above it, and row n holds the binomial coefficients — the coefficients of (a+b)ⁿ. The triangle hides the Fibonacci numbers, powers of two, and the Sierpinski pattern.Perfect Number CheckerTest whether a number is perfect, abundant or deficient from the sum of its proper divisors, with the Euclid-Euler theorem explained.Power Series CalculatorExpand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.Recurrence relation solverGenerate the sequence of any linear recurrence a(n) = c₁·a(n−1) + … + cₖ·a(n−k). Enter the coefficients and the matching initial values, and it unrolls the sequence term by term — Fibonacci, Lucas, Pell and any custom order-k relation you define, with decimals and negatives supported.Series Convergence Test CalculatorPick a series and see which test settles it — p-series, ratio, root, Leibniz, Cauchy condensation, limit comparison — with the limit it produces and whether the convergence is absolute or conditional.Square numbers listGenerate a list of perfect squares — n² for consecutive integers from any starting point. Choose how many and where to begin, and it returns the squares and their sum with exact big-integer arithmetic, handy for spotting patterns and differences.Triangular number calculatorCompute the nth triangular number (1 + 2 + … + n).