Track / Stage 4 / 156 of 340 #416 of 1964

Problem 416

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

Arrange the numbers of Pascal's Triangle aligned to the far left as follows:

The numbers in the diagonal direction connected by arrows are referred to as a diagonal, for example, the first diagonal is “ $1 \leftarrow 1$ ”, the second diagonal is “ $1 \longleftrightarrow \rightarrow 2$ ”, and the third diagonal is “ $1 \longleftrightarrow 3 \longleftrightarrow \rightarrow 1$ ”.
What is the sum of all the numbers on the 13th diagonal?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

【Analysis】Find the pattern
The sums are $2,3,5,8,13,21,34,55,89,144,233,377,610 ; \ldots .$ . (Notice that each subsequent number is the sum of the previous two)
The thirteenth sum is 610.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.