Draw Pascal Triangle

Pascal’s triangle is the triangular array where every interior entry is the sum of the two entries directly above it. Row n (starting from row 0) holds the binomial coefficients C(n, 0), C(n, 1), …, C(n, n) – the coefficients of (a + b) to the n. It is named after Blaise Pascal (Traité du triangle arithmétique, 1654), but earlier descriptions appear in works by Halayudha (India, 10th century), Omar Khayyam (Persia, 11th century), Yang Hui (China, 13th century, where it is still called Yang Hui’s Triangle), and Petrus Apianus (Germany, 1527).

Yes. The tool uses JavaScript’s BigInt for every cell from row 0 onward, so all coefficients are exact integers regardless of size. The largest value at R = 100 is C(99, 49) which has 29 digits – JavaScript Number would have lost precision past row 53 (Number.MAX_SAFE_INTEGER is 2 to the 53 minus 1). An earlier version of this FAQ claimed “Number type up to row 30, BigInt for larger” – that was wrong; the code has always used BigInt throughout. Now corrected.

Colour each cell by C(n, k) mod 2 – odd cells filled, even cells empty. By Lucas’s theorem (1878), C(n, k) is odd if and only if every binary digit of k is at most the corresponding binary digit of n; otherwise it is even. The resulting pattern is a discrete approximation of the Sierpinski triangle, with the fractal becoming visually convincing from about row 16 (2^4) and getting sharper at rows 32, 64, 128 (powers of 2). It is the same fractal you get from the Sierpinski triangle’s classical IFS construction.

The sum of row n is exactly 2 to the n. This follows from setting a = b = 1 in the binomial theorem: (1 + 1) to the n equals the sum of C(n, k) for k from 0 to n equals 2 to the n. Combinatorially it counts all subsets of an n-element set (each element is either in the subset or out). The stats line at the top of the output shows this for the last row drawn.

Sum along the “shallow diagonals” – paths that go one cell right and one cell up-left at each step. The diagonals’ sums are 1, 1, 2, 3, 5, 8, 13, 21, … the Fibonacci numbers. This identity is sum_{k=0}^{floor(n/2)} C(n-k, k) = F(n+1). It is most visible in the Numbers mode for R ≥ 10.

Rows 0 through 4 read off as 11^0 through 11^4 directly: 1, 11, 121, 1331, 14641. From row 5 onward the interior values exceed 9, so the digits “carry” into neighbouring positions and the simple concatenation breaks. The pattern is real but trivial – it is just base-10 digit expansion of 11 to the n.

Diagonal 0 (the outer edge) is all 1s. Diagonal 1 is the natural numbers 1, 2, 3, 4, … Diagonal 2 is the triangular numbers 1, 3, 6, 10, 15, … Diagonal 3 is the tetrahedral numbers 1, 4, 10, 20, 35, … Diagonal k holds the n-th figurate number for the k-simplex: C(n, k) for n = k, k+1, k+2, …

No. The page loads three static files (HTML, CSS, JS) and then computes everything in your browser using BigInt arithmetic. You can disconnect from the internet after the page loads and the tool keeps working. No analytics, no telemetry, no cookies.

Yes – free, unlimited, no signup, no watermark. Use the output in lectures, assignments, papers, or blog posts freely. Attribution to is appreciated but not required.