BYTETOOLS

Pascal's Triangle Generator

Generate Pascal's triangle to any number of rows with exact big integers, look up any C(n, k), and highlight the Sierpiński and diagonal patterns.

252
C(10, 5)
1,024
Row 10 sum = 2^10
8
Rows shown

Pascal's triangle

1
11
121
1331
14641
15101051
1615201561
172135352171

Every entry is the sum of the two directly above it: C(n, k) = C(n−1, k−1) + C(n−1, k). All values are computed as exact big integers, so the wide rows stay correct rather than rounding off.

What is the Pascal's Triangle Generator?

The ByteTools Pascal's Triangle Generator builds the triangle row by row using the recurrence C(n, k) = C(n−1, k−1) + C(n−1, k), with every entry computed as an exact big integer so the wide rows stay correct instead of rounding off into scientific notation.

  • Exact big-integer entries — no rounding on the wide rows
  • Single-entry C(n, k) lookup for n up to 1000
  • Row sum 2ⁿ shown alongside the lookup
  • Odd/even highlight that reveals the Sierpiński triangle
  • Triangular and tetrahedral number diagonals highlighted
  • Hockey-stick identity demonstrated on the triangle

How to use the Pascal's Triangle Generator

  1. 1

    Set how many rows of the triangle you want, up to 30.

  2. 2

    Choose a highlight to reveal the Sierpiński pattern, a number diagonal or the hockey stick.

  3. 3

    Enter n and k in the lookup boxes to get a single binomial coefficient and its row sum.

  4. 4

    Tick the plain-text option if you want a version that is easy to copy or print.

  5. 5

    Click Copy triangle to take the whole thing with you.

About the Pascal's Triangle Generator

The ByteTools Pascal's Triangle Generator builds the triangle row by row using the recurrence C(n, k) = C(n−1, k−1) + C(n−1, k), with every entry computed as an exact big integer so the wide rows stay correct instead of rounding off into scientific notation.

Alongside the triangle you get a single-entry lookup for any C(n, k) up to n = 1000, the row sum 2ⁿ, and highlight modes that reveal the patterns hiding in it: the odd entries trace out the Sierpiński triangle, the third and fourth diagonals hold the triangular and tetrahedral numbers, and the hockey-stick highlight demonstrates that identity on the triangle itself.

A plain-text view makes the triangle easy to copy or print for classroom use. Everything runs in your browser and nothing is uploaded.

Frequently asked questions

What is Pascal's triangle used for?

Its entries are the binomial coefficients, so it gives the coefficients when you expand (a + b)ⁿ. It also answers combination questions — C(n, k) is the number of ways to choose k items from n — and encodes the triangular and tetrahedral numbers along its diagonals.

How do you calculate the entries in Pascal's triangle?

Each entry is the sum of the two directly above it, starting from a single 1 at the top and 1s down both edges. Equivalently, the entry in row n and position k is the binomial coefficient C(n, k) = n! ÷ (k!(n−k)!).

What is the hockey stick identity?

Adding a straight diagonal run of entries always equals the entry one row down and one place across: C(k,k) + C(k+1,k) + … + C(n,k) = C(n+1,k+1). The highlighted cells form a hockey-stick shape, which is where the name comes from.

Why does Pascal's triangle make the Sierpiński triangle?

Colouring only the odd entries leaves a fractal pattern of triangular holes that repeats at every scale. It happens because whether C(n, k) is odd depends on the binary digits of n and k, and that rule is self-similar.

What is the sum of each row in Pascal's triangle?

Row n adds up to 2ⁿ. Row 0 sums to 1, row 3 to 8, row 10 to 1,024. It follows from setting a = b = 1 in the binomial expansion of (a + b)ⁿ.

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