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Tetrahedral Number Calculator

Work out tetrahedral number instantly with clear inputs, formula shown and shareable results.

About Tetrahedral Number Calculator

Tetrahedral numbers stack triangular layers into a pyramid, giving Teₙ = n(n + 1)(n + 2)/6 — the running total of the triangular numbers. They also form the third diagonal of Pascal's triangle.

Formula

Tetrahedral number

Teₙ = n(n + 1)(n + 2) / 6 = C(n + 2, 3)

Worked example

With the default values below, Tetrahedral Number Calculator returns nth tetrahedral number of 220. Change any input above and every figure here is recalculated in your browser.

Inputs used

Index naccepts 1–1,000,000
10
How many terms to listaccepts 1–20
6

Results

nth tetrahedral number
220
Next tetrahedral number
286
Size of the base layer
55
First terms of the sequence
1, 4, 10, 20, 35, 56

Step by step

  1. 1Values used
  2. 2Tetrahedral number
  3. 3nth tetrahedral number= 220
  4. 4Next tetrahedral number= 286
  5. 5Size of the base layer= 55
  6. 6First terms of the sequence= 1, 4, 10, 20, 35, 56

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the 10th tetrahedral number?

10 × 11 × 12 / 6 = 220, the number of cannonballs in a ten-layer pyramid.

Which tetrahedral numbers are also square?

Only 1, 4 and 19600 — a result proved in the 1870s.

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