Tetrahedral number calculator

The tetrahedral number calculator returns the n-th tetrahedral number for any positive integer index n. By definition that value is the sum of the first n triangular numbers, and it has the closed form n times n plus one times n plus two, all divided by six — the same quantity as the binomial coefficient choose n plus two, three. The first few terms are one, four, ten, twenty, thirty-five, fifty-six, eighty-four, one hundred twenty, one hundred sixty-five, and two hundred twenty. This capability is a pure parse engine: it multiplies and divides exact integers with BigInt, never approximates with floating-point roots, never samples randomness, never reads the clock, and never calls the network.

What a tetrahedral number is and how Te_n is defined

A tetrahedral number counts how many equal spheres you can stack in a triangular pyramid with n layers. The bottom layer is a triangular number T_n = n(n+1)/2, the layer above is T_(n-1), and so on down to a single sphere at the apex. Summing those layers gives the n-th tetrahedral number: Te_n equals the sum from k equals one to n of T_k.

That sum collapses to the compact formula Te_n = n(n+1)(n+2)/6, which is also the binomial coefficient C(n+2, 3). For n equals one the result is one; for two it is four; for three it is ten; for four it is twenty; for five it is thirty-five; for ten it is two hundred twenty.

The formula is exact for every positive integer n: among any three consecutive integers n, n+1, and n+2, at least one is divisible by two and at least one by three, so the product is always divisible by six and the result is a whole number. Valid positive indices always succeed with a deterministic tetrahedral value; there is no probabilistic path and no floating-point step that could round a near-miss.

How inputs are read and how the closed form stays exact

Optional leading plus signs are allowed; embedded spaces, hex prefixes, and exponential notation are rejected so a bad spreadsheet cell does not become a wrong pyramid count. Values below one are rejected with an explicit message: the tetrahedral sequence here starts at n equals one, matching the usual combinatorial and geometric definitions. Internally the product n times n plus one times n plus two is formed with BigInt, then divided by six, so intermediate products never lose low-order bits the way a double multiply might for large n.

Pair the output with triangular-number tables when you want to show the partial sums explicitly, or with binomial-coefficient tools when you prefer the C(n+2, 3) view of the same integer.

Where tetrahedral numbers fit in study, puzzles, and automation

Students meet tetrahedral numbers when learning figurate sequences, binomial coefficients, and three-dimensional stacking models after they already know triangular numbers. Teachers can generate answer keys for worksheets without trusting opaque calculator apps that hide floating-point division. Puzzle and game designers use tetrahedral counts for cannonball piles, tetrahedron resource costs, and level progression curves that grow as a cubic polynomial.

Developers use the same contract to validate sample data for number-theory libraries, to seed property tests that need known figurate values, or to pre-check that a configuration size matches a tetrahedral count before a simulation step that assumes stacked triangular layers. Contest practice often needs a fast offline source of Te_n so solutions can be checked without downloading OEIS tables or hard-coding long lists. Data pipelines that size batch jobs or pyramid-shaped caches sometimes need a closed-form item count from a depth parameter.

Pair it with square-number or Lucas tools when you need other classical sequences, and with combinatorial helpers when you need binomials beyond C(n+2, 3).