Arithmetic Slices II  Subsequence
The problem for the number of all arithmetic subsequences in the given integer array nums
. A sequence is called arithmetic if it consists of at least three elements and the difference between any two consecutive elements is the same.
We can solve this problem by using a dynamic programming approach.
 Initialize a DP Table: We will use a list of dictionaries, where
dp[i][diff]
represents the number of arithmetic subsequences that end at indexi
with a common difference ofdiff
.  Iterate Through the Array: We will iterate through the array, and for each pair of elements, calculate the difference and update the DP table.
 Count the Arithmetic Sequences: While updating the DP table, we also keep track of the count of valid arithmetic sequences.
Here’s the code:


The time complexity of this code is (O(n^2)), and the space complexity is also (O(n^2)), where (n) is the length of the nums
array.
Example Explanation:
 For
nums = [2,4,6,8,10]
, the output will be7
, as there are seven arithmetic subsequences:[2,4,6]
,[4,6,8]
,[6,8,10]
,[2,4,6,8]
,[4,6,8,10]
,[2,4,6,8,10]
,[2,6,10]
.  For
nums = [7,7,7,7,7]
, the output will be16
, as any subsequence of this array is arithmetic.


Identifying Problem Isomorphism
“Arithmetic Slices II  Subsequence” asks you to count the number of arithmetic subsequences in an array, a bit more complicated than simply identifying them.
A simpler problem is “Arithmetic Slices”, which asks you to count the number of arithmetic subarrays. This problem doesn’t require handling the additional complexity of subsequences, which do not have to contain adjacent elements.
An approximately isomorphic problem is “Number of Longest Increasing Subsequence”. It’s not exactly the same, but similar in the sense that you’re looking for certain types of subsequences.
A more complex problem is “Distinct Subsequences II”. It requires you to count subsequences under different conditions, and the sequences don’t necessarily need to be arithmetic.
Here is the ordered list of problems from simplest to most complex:
 “Arithmetic Slices”  Count the number of arithmetic subarrays.
 “Arithmetic Slices II  Subsequence”  Count the number of arithmetic subsequences.
 “Number of Longest Increasing Subsequence”  Count subsequences that have a different condition.
 “Distinct Subsequences II”  Count subsequences under more complex conditions.
“446. Arithmetic Slices II  Subsequence” is about finding arithmetic subsequences in an array. It involves understanding of dynamic programming (DP) and hashing. Here are 10 problems to prepare for it:
413. Arithmetic Slices: This is a simpler version of the problem where you need to find arithmetic subarrays instead of subsequences.
300. Longest Increasing Subsequence: This problem is a classic dynamic programming problem about subsequences.
673. Number of Longest Increasing Subsequence: This is a more complex version of problem 300 where you need to count the number of longest increasing subsequences.
646. Maximum Length of Pair Chain: This problem is also about finding the longest chain (subsequence) with certain properties.
152. Maximum Product Subarray: This problem asks for the contiguous subarray which has the largest product, which will give you good practice working with subarrays.
198. House Robber: A classic dynamic programming problem that helps build a foundation for understanding how to approach dynamic programming problems.
560. Subarray Sum Equals K: This problem will help you understand how to handle the cumulative sum concept which is useful for this problem.
718. Maximum Length of Repeated Subarray: This problem can help you get familiar with working with arrays and finding subsequences.
494. Target Sum: This problem is about finding all the subsets of an array that sum to a particular target, which can help you understand how to traverse all subsequences of an array.
53. Maximum Subarray: This problem is a basic one in dealing with subarrays, and will help you get comfortable with the concept.
Clarification Questions
What are the clarification questions we can ask about this problem?


The task is to count the number of arithmetic subsequences in an array. This is a Arithmetic Subsequence Counting Problem.
Language Agnostic Coding Drills
This problem is about counting all the arithmetic sequences in a list of numbers (also known as arithmetic slices). An arithmetic sequence is a sequence of numbers in which the difference between consecutive numbers is constant.
Drills:
Understanding Lists, Arrays or Vectors: This is the fundamental data structure used in this problem. The input is a list of integers, and the goal is to find patterns within this list.
Understanding Loops and Nested Loops: The main logic of the solution involves two nested loops that iterate through the list. Understanding how to use loops to iterate through a list, and particularly how to use one loop inside another, is crucial.
Understanding Arithmetic Sequences: An arithmetic sequence is a sequence of numbers such that the difference of any two successive members is a constant. In this problem, you’re asked to find all such sequences within the given list.
Understanding Dictionaries or Hash Maps: Dictionaries are used to keep track of the arithmetic slices found so far. This is an important part of the solution, as it allows us to effectively count the number of slices.
Understanding Dynamic Programming: The solution to this problem involves a dynamic programming approach, where you store solutions to subproblems (in this case, the number of arithmetic slices ending at each position in the list) and use them to solve the overall problem.
Mathematics: This problem requires a clear understanding of arithmetic progression and summation of series. You need to calculate the difference between elements and check if it’s consistent.
Problem Solving: This problem requires building up a solution incrementally, understanding the relationship between a smaller problem (the number of arithmetic slices ending at a particular position) and the overall problem.
Approach to the solution:
 Create a list of dictionaries to track the number of arithmetic slices ending at each position in the list.
 Iterate through the list twice using nested loops. In the inner loop, calculate the difference between the current element of the outer loop and the current element of the inner loop.
 Update the dictionary at the outer loop’s current position by incrementing the count for this difference by one, and also add the count for this difference at the inner loop’s current position.
 Increase the total count of arithmetic slices by the count for this difference at the inner loop’s current position.
 Return the total count of arithmetic slices.
Targeted Drills in Python
Coding Drill 1: Understanding Lists, Arrays or Vectors
Create a list and perform various operations like appending elements, accessing elements, and iterating through the list.


Coding Drill 2: Understanding Loops and Nested Loops
Use nested loops to iterate through a list of lists.


Coding Drill 3: Understanding Dictionaries or Hash Maps
Create a dictionary, add keyvalue pairs to it, and iterate through it.


Coding Drill 4: Understanding Dynamic Programming
Implement a dynamic programming solution to the Fibonacci sequence.


Coding Drill 5: Mathematics  Arithmetic Progression
Write a function to generate an arithmetic progression.


Problem Specific Drill:
This problem involves identifying and counting arithmetic slices within a list. The following drill builds on the skills practiced in the previous drills and applies them to a simpler version of the problem: identifying and counting arithmetic slices of exactly length 3 within a list.

