More relations between sequences

Three more surprising links between sequences.

1. Powers of 2. Keep a running total of 1, 2, 4, 8, 16, …: 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15. Add 1 to each: 2, 4, 8, 16: the powers of 2 again! Each sum is one less than the next power of 2. Why? Put an extra 1 at the front and watch it grow: 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16. Each time the total so far doubles.

2. Triangular to hexagonal. Multiply a triangular number by 6 and add 1: 6 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, 6 × 10 + 1 = 61. These are the hexagonal numbers! A hexagon of dots splits into one centre dot and six equal triangles around it. In the hexagon of 19 dots, each piece is a triangle of 3 dots: 19 = 6 × 3 + 1.

3. Hexagonal to cubes. Keep a running total of 1, 7, 19, 37, 61, …: 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64. These are the cubes: 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, 4 × 4 × 4! Look at a cube of 27 small cubes from one corner, so you see three faces. You see 19 small cubes, in a shape like a hexagon, and hidden behind them is a smaller cube of 8. So 27 = 19 + 8 = 19 + 7 + 1.