0086EasyMathDynamic ProgrammingRecursionMemoization

Cascade Register Reading

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Trains the technique from

This is an original problem, written from a brief that listed the technique, the difficulty, the topics, the function shape and the input bounds — none of that problem's wording, examples, hints or editorials. The link is there so you can map your practice onto the standard set.

Same function shape, different story and different numbers.

A ramp generator on a test rig publishes a numbered stream of readings. Reading 0 comes out as 0 and reading 1 comes out as 1. From then on the register produces reading k by totalling the two readings it published immediately beforehand, that is reading k - 1 and reading k - 2.

The stream therefore opens 0, 1, 1, 2, 3, 5, 8 and carries on.

Given an index n, return reading n.

Examples

Example 1

Input: n = 0
Output: 0

The stream opens on 0, and no totalling is needed.

Example 2

Input: n = 7
Output: 13

Readings 5 and 6 come out as 5 and 8, and totalling them gives reading 7.

Example 3

Input: n = 16
Output: 987

Working up the stream one reading at a time reaches 987 at index 16.

Constraints

0 <= n <= 30

The signature

The editor is preloaded with this. It matches the parent problem's shape, so a solution that works here transfers to a judge unchanged.

Python

def cascade_reading(n: int) -> int:

Java

public int cascadeReading(int n)

The intended approach

Time O(n)Space O(1)

Reading `k` only ever depends on the two readings behind it, so nothing older has to be kept. Hold that pair in two variables, roll it forward once per index up to `n`, and hand back the newer of the two. Index `0` is answered directly because the stream has no earlier pair there.

Pythonentry point: cascade_reading

def cascade_reading(n):
    if n == 0:
        return 0
    earlier, latest = 0, 1
    for _ in range(n - 1):
        earlier, latest = latest, earlier + latest
    return latest

The obvious version

Unwind the definition top down with plain recursion and no cache: an index below two answers itself, otherwise total the two calls beneath it. Exponentially many calls, but it mirrors the wording of the rule exactly.

This is not filler. It was written independently of the fast solution and the two were run against thousands of random inputs to confirm they agree — which is how the answer above is known to be right. Read it when the intended approach feels like a rabbit pulled out of a hat.

Python

def bf_cascade_reading(n):
    if n < 2:
        return n
    return bf_cascade_reading(n - 1) + bf_cascade_reading(n - 2)

How this problem was checked

Reference solution executed against 10 test cases — 10 passed
3 worked examples confirmed by running the code
0 eight-word sequences shared with the parent problem's text
Agrees with an independent brute-force implementation on random inputs
Test set proven to catch deliberately broken solutions
Reviewed by a separate quality pass
Function arity matches the parent problem's

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