Interactive Pascal triangle with pattern highlighting: Fibonacci diagonals, powers of 2, Sierpinski coloring, and binomial coefficients.
This tool builds Pascal's triangle row by row using the addition rule (each interior entry is the sum of the two entries above it) and layers optional highlighting on top: odd/even shading that reveals the Sierpinski gasket, diagonal readouts that sum to Fibonacci numbers, and row-sum readouts that equal powers of two. The numbers underneath never change; only the coloring and annotations do. What trips people up most is indexing - both rows and positions within a row start at 0, so 'the 5th number in row 6' visually is not C(6,5); it's whichever 0-indexed entry you actually meant.
A concrete input and expected output from the current implementation.
Input
Set rows = 6 (rows 0 through 5), binomial coefficient labels on
Expected output
Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1
Each row starts and ends in 1; every interior value is the sum of the two values diagonally above it (e.g. row 5's 10 = 4+6 from row 4), which is equivalent to C(n,k) for row n, position k.
Rows are generated with Pascal's recurrence rather than by evaluating factorials directly: each row begins and ends with 1, and every interior cell is the sum of the two cells diagonally above it in the previous row. This is the standard way to produce C(n,k) values and keeps intermediate arithmetic simple even though the final numbers themselves can still grow very large.
Toggleable overlays recolor or annotate the same grid without altering the underlying values. An odd/even (mod-2) overlay shades cells to reveal the Sierpinski triangle fractal; a shallow-diagonal readout sums entries along each diagonal to show consecutive Fibonacci numbers; a row-sum readout shows that each row totals 2 raised to that row's index.
Based on the tool's stated features, the Sierpinski overlay is the classic mod-2 parity coloring, and there's no mention of a configurable modulus for mod-3 or higher. Treat those patterns as outside this tool's documented capability unless you find an explicit modulus control in the interface.
This isn't documented by the tool itself. As a general rule, if it relies on standard double-precision numbers, exact integer accuracy holds up to about 9 quadrillion (2^53), which central entries approach somewhere in the high 50s of rows - beyond that, double-check any very large displayed value rather than assuming full precision.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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