What is the next number is this sequence? 1, 4, 10, 20, 35, 56, ?

This sequence is a list of the tetrahedral numbers, and the next value is 84. These numbers represent the total count of objects arranged in a three-sided pyramid, known as a tetrahedron. The first number, 1, is a single point. The second number, 4, corresponds to a tetrahedron with two layers (1 on the top and 3 on the base). The sequence continues by adding another triangular layer to the base for each new number.

To find the next number in the sequence, one can look at the differences between consecutive terms. The difference between 4 and 1 is 3, between 10 and 4 is 6, and between 20 and 10 is 10. These differences are the triangular numbers. Following this pattern, the difference between 56 and 35 is 21. The next triangular number after 21 is 28, which is then added to 56 to arrive at 84. Each tetrahedral number is the sum of the preceding triangular numbers.

These fascinating numbers can be visualized as a stack of spheres, like billiard balls in a triangular frame, with each layer representing a triangular number. They also appear in Pascal's triangle, in the fourth diagonal position. This connection to other mathematical concepts highlights the interconnectedness of different numerical patterns and their representations in geometry and other fields of study.