Pascal's Triangle

Every interior number is the sum of its two upper neighbors from the row above, while the edges are fixed at 1. Generating row by row lets each stage directly consume the values computed in the immediately preceding row without recalculating combinations from scratch.

Full solution

Iterative row construction: each row starts as all 1s and every interior cell is the sum of the two cells above it in the previous row. No factorials, so nothing overflows and the edges never index out of bounds.

def generate(n: int) -> list[list[int]]:
    rows: list[list[int]] = []
    for r in range(n):
        row = [1] * (r + 1)
        # Interior cells sum the two cells above; edges stay 1.
        for j in range(1, r):
            row[j] = rows[r - 1][j - 1] + rows[r - 1][j]
        rows.append(row)
    return rows

function generate(n) {
  const rows = [];
  for (let r = 0; r < n; r++) {
    const row = new Array(r + 1).fill(1);
    // Interior cells sum the two cells above; edges stay 1.
    for (let j = 1; j < r; j++) {
      row[j] = rows[r - 1][j - 1] + rows[r - 1][j];
    }
    rows.push(row);
  }
  return rows;
}

The theory behind it

Arrays — the ground this problem stands on. All Arrays problems

What Arrays is

An array is a row of fixed boxes laid side by side in computer memory, like numbered lockers in a hallway. Because each box occupies identical space and sits directly next to its neighbors, jumping to locker zero or locker ten thousand takes the exact same tiny fraction of time. Every box holds an item of the same type, addressed by an offset number called an index.

When to reach for it

Reach for an array when items arrive in a known sequence and need immediate retrieval by position number. Problems asking for running totals, prefix accumulations, cyclic rotations, or in-place rearrangements signal array mechanics. Whenever constraints require constant-time random lookups or contiguous cache scans across fixed collections, a flat sequence is the default container.

How the pattern works

Visualize a tape with zero-indexed slots stretching from start to end. Keep track of write and read cursors when modifying contents without allocating helper buffers. For running computations, maintain an invariant such as having processed all elements left of the current index while pending elements wait to the right. When modifying entries in place, consider scanning backwards from the end so unread data is not overwritten.