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Triangular Numbers Calculator

Calculate nth triangular numbers, test whether a value is triangular, and preview the sequence with exact integer arithmetic.

Last Updated: May 2026

Figurate numbers

Calculate and test triangular numbers

Triangular numbers count dots arranged in an equilateral triangle. Compute T_n, check whether a value is triangular, and preview the sequence with exact integer arithmetic.

Calculator mode

Used for T_n mode. n must be a nonnegative integer.

Used for check mode. Enter a nonnegative integer.

Show the first 1 to 80 triangular numbers.

Examples

Triangular number checks

ItemFormulaResult
Triangular formulaT_n = n(n + 1) / 2T_10 = 55
Dot pattern1 + 2 + ... + nSum through 10
Previous triangularT_(n - 1)45
Next triangularT_(n + 1)66
Triangular test8x + 1 must be a perfect squarePasses

Sequence preview

nFormulaT_n
11 x 2 / 21
22 x 3 / 23
33 x 4 / 26
44 x 5 / 210
55 x 6 / 215
66 x 7 / 221
77 x 8 / 228
88 x 9 / 236
99 x 10 / 245
1010 x 11 / 255

Integer Sequence Notice

This calculator works with nonnegative integer triangular numbers. Decimal, fraction, and symbolic sequence inputs are outside this page.

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Methodology & Updates

Page updated May 2026. Trust-critical pages are reviewed when official rates or rules change. Evergreen calculator guides are checked on a recurring quarterly or annual cycle depending on topic volatility.

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How to Use the Triangular Numbers Calculator

Choose whether to calculate T_n from an index or check whether a value is triangular. Enter nonnegative integers only.

Review the main result, the n(n + 1) / 2 formula, the 8x + 1 perfect-square check, and the sequence preview.

  1. Step 1: Choose a mode

    Use Find T_n to calculate from an index, or Check a value to test an integer.

  2. Step 2: Enter the index or value

    Use nonnegative integers such as 10, 100, 1225, or 1000000.

  3. Step 3: Read the formula result

    The calculator shows T_n = n(n + 1) / 2 and neighboring triangular numbers.

  4. Step 4: Use the square test

    For value checks, 8x + 1 must be a perfect square for x to be triangular.

How This Triangular Numbers Calculator Works

To calculate the nth triangular number, the tool uses T_n = n(n + 1) / 2. This is the same as adding 1 + 2 + ... + n.

To check a value x, the tool computes 8x + 1 and tests whether it is a perfect square. If it is, the triangular index is (sqrt(8x + 1) - 1) / 2.

Calculations use exact integer arithmetic, and the sequence preview lists the first triangular numbers up to the count you choose.

Triangular Numbers Guide

Triangular Number Rules

ConceptFormula or ExampleMeaning
DefinitionT_n = 1 + 2 + ... + nThe sum of the first n positive integers.
Closed formT_n = n(n + 1) / 2Fast formula for the nth triangular number.
First values1, 3, 6, 10, 15The dot pattern grows by one more dot each row.
Zero indexT_0 = 0Useful in some sequence and combinatorics contexts.
Triangular test8x + 1 is an odd squareA value x is triangular exactly when this condition holds.

Examples

InputWorkResult
T_1010 x 11 / 255
T_100100 x 101 / 25,050
1,2258 x 1,225 + 1 = 9,801 = 99^2Triangular, n = 49
1,0008 x 1,000 + 1 = 8,001Not triangular
T_1,000,0001,000,000 x 1,000,001 / 2500,000,500,000

Why T_n Uses n(n + 1) / 2

Pair the sequence 1 + 2 + ... + n with the same sequence reversed. Each pair adds to n + 1, and there are n pairs across the doubled sum. Dividing by 2 gives n(n + 1) / 2.

Keep the research moving with Consecutive Integers Calculator, Integer Calculator, Perfect Square Calculator, and Sum of Products Calculator.

Frequently Asked Questions

A triangular number is the sum of the first n positive integers, written T_n = n(n + 1) / 2.

They count dots that can be arranged into an equilateral triangle with n dots on each side.

A nonnegative integer x is triangular when 8x + 1 is a perfect odd square. Then n = (sqrt(8x + 1) - 1) / 2.

Many sequence definitions include T_0 = 0. This calculator supports n = 0 and nonnegative values.

Yes. The calculator uses exact BigInt arithmetic for integer calculations, so large triangular numbers are not rounded.

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Sources & References

  1. 1.Wolfram MathWorld - Triangular Number(Accessed May 2026)
  2. 2.Wikipedia - Triangular number(Accessed May 2026)
  3. 3.OpenStax Algebra and Trigonometry - Sequences(Accessed May 2026)