Sequence Tools

MiniWebtool’s Arithmetic Sequence Calculator finds the nth term and sum of an arithmetic progression from its first term, common difference, and term count. Worked substitutions and a visual representation explain the constant-step pattern.

MiniWebtool’s Catalan Number Generator calculates Cn and illustrates why it counts objects such as balanced parentheses and Dyck paths. Enter an index to inspect the exact value and its derivation.

MiniWebtool’s Collatz Conjecture Calculator traces a positive integer by halving even terms and applying 3n + 1 to odd terms. Inspect the trajectory and statistics for your input; the conjecture remains unproved for all positive integers.

Generate a range of perfect cubes with MiniWebtool’s Cube Numbers List. Choose a starting value and count to explore the sequence, accompanying statistics, visualizations, and available export formats.

Generate a list of consecutive prime numbers with MiniWebtool’s First N Prime Numbers tool. Enter a number from 1 to 10,000 to get the list, identify twin primes and view the gaps between each prime.

Trace a sequence whose terms share a constant ratio with MiniWebtool’s Geometric Sequence Calculator. It calculates a selected term and finite sum, and gives an infinite sum when the ratio allows the series to converge.

Repeatedly square a number’s digits and add them: does the sequence reach one? MiniWebtool’s Happy Number Calculator traces that journey, identifies happy numbers or loops, and lets you explore happy numbers within a range.

Choose how far to follow the Fibonacci sequence in MiniWebtool’s List of Fibonacci Numbers. Generate terms by count, maximum value, or index range, with a spiral diagram and a view of the sequence’s golden-ratio relationship.

List the prime numbers up to a chosen limit of 10,000 with MiniWebtool’s List of Prime Numbers. The generated sequence and summary statistics provide a reference for exploring factors, divisibility, and prime-number patterns.

MiniWebtool’s Magic Square Generator constructs number grids whose rows, columns, and main diagonals share a sum. Inspect the construction steps, magic constant, and mathematical properties for the selected supported order.

MiniWebtool’s Partition Function Calculator counts integer partitions p(n) for n from zero to 500. Compare unrestricted, distinct-part, and odd-part counts, and explore individual partitions with Young diagrams when n is at most 15.

Explore binomial coefficients and numerical patterns in MiniWebtool’s Pascal's Triangle Generator. Generate up to 30 rows and highlight relationships with powers of two, Fibonacci numbers, and the Sierpinski pattern.

Expand and evaluate a finite product in MiniWebtool’s Product Notation Calculator (Pi Notation). The expression and index bounds produce individual factors and running products, making Π notation’s multiplication process visible.

Move from a recursive definition to an explicit formula with MiniWebtool’s Recurrence Relation Solver. It handles linear homogeneous recurrences with constant coefficients and initial values, returning a closed form and up to sixty terms.

Evaluate sigma (Σ) summation notation expressions with MiniWebtool’s Sigma Notation Calculator, including step-by-step term expansion, running total visualization, and convergence analysis. It supports polynomial, exponential, trigonometric, and custom expressions.

Explore perfect squares with MiniWebtool’s Square Numbers List. Generate a chosen number of squares, find them within a range, or test whether one value is a perfect square.

Find every twin prime pair (primes p and p+2) up to any limit you choose with MiniWebtool’s Twin Prime Finder. Get the complete list, totals, density per decade, the predicted Hardy-Littlewood count, largest pair found, and an interactive visualization in one place.