Instead of storing the entire Padovan sequence in a list, you can optimize the space complexity by just storing the last three elements of the sequence. This way, you can calculate the next element based on these three elements without storing the entire sequence.

Write a Python program using functions and recursion to generate and display the first 15 numbers of the Padovan sequence. The Padovan sequence begins with the first three terms as 1, 1, and 1, and each subsequent term is the sum of the previous two terms. For example, the first few terms of the Padovan sequence are: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37,.Note: The Padovan sequence is a sequence of numbers similar to the Fibonacci sequence, where each term is the sum of the previous two terms.

Given a number n, find the nth number in the Padovan Sequence.A Padovan Sequence is a sequence which is represented by the following recurrence relationP(n) = P(n-2) + P(n-3)P(0) = P(1) = P(2) = 1Note: Since the output may be too large, compute the answer modulo 10^9+7.

Given an array, find the length of the longest subsequence whose elements can be re-arranged in a strictly increasing contiguous order. The difference between 2 adjacent elements in the subsequence, after re-arrangement, should be exactly 1.Input FormatThe first line of input contains T - the number of test cases. It's followed by 2T lines. The first line of each test case contains N - size of the array. The next line contains N integers - the elements of the array.Output FormatFor each test case, print the length of the longest subsequence, separated by a new line.Constraints1 <= T <= 10001 <= N <= 10000-100000 <= ar[i] <= 100000ExampleInput3821 -22 -22 5 -31 -24 5 -231018 -33 31 33 30 -14 32 30 16 1766 3 8 5 2 5Output342ExplanationTest Case 1Subsequence is: -22, -24, -23.Test Case 2Subsequence is: 31, 33, 30, 32.Test Case 3Subsequence is: 6, 5 or 3, 2.

longest consecutive elementsYou need to write a function that takes an unsorted array of integers as input and returns the length of the longest consecutive elements sequence in the array. A consecutive elements sequence is a subarray where the elements are adjacent in value, such as [1, 2, 3, 4] or [5, 6, 7, 8]. The order of the elements in the input array does not matter.For example, given the input array [100, 4, 200, 1, 3, 2], the longest consecutive elements sequence is [1, 2, 3, 4], which has a length of 4. Therefore, your function should return 4.In this case, there are no consecutive elements in the array, so the length of the longest consecutive element sequence is 1.Testcase:Input11  // no of elements of array8 10 12 14 1 2 3 4 5 6 7  // array elementsOutput:8

You are given a sequence [a1,…,an][𝑎1,…,𝑎𝑛] where each element ai𝑎𝑖 is either 00 or 11. You can apply several (possibly zero) operations to the sequence. In each operation, you select two integers 1≤l≤r≤|a|1≤𝑙≤𝑟≤|𝑎| (where |a||𝑎| is the current length of a𝑎) and replace [al,…,ar][𝑎𝑙,…,𝑎𝑟] with a single element x𝑥, where x𝑥 is the majority of [al,…,ar][𝑎𝑙,…,𝑎𝑟].Here, the majority of a sequence consisting of 00 and 11 is defined as follows: suppose there are c0𝑐0 zeros and c1𝑐1 ones in the sequence, respectively.If c0≥c1𝑐0≥𝑐1, the majority is 00.If c0<c1𝑐0<𝑐1, the majority is 11.For example, suppose a=[1,0,0,0,1,1]𝑎=[1,0,0,0,1,1]. If we select l=1,r=2𝑙=1,𝑟=2, the resulting sequence will be [0,0,0,1,1][0,0,0,1,1]. If we select l=4,r=6𝑙=4,𝑟=6, the resulting sequence will be [1,0,0,1][1,0,0,1].Determine if you can make a=[1]𝑎=[1] with a finite number of operations.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤4⋅1041≤𝑡≤4⋅104). Description of the test cases follows.The first line of each testcase contains one integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105).The second line of each testcase contains a string consisting of 00 and 11, describing the sequence a𝑎.It's guaranteed that the sum of n𝑛 over all testcases does not exceed 2⋅1052⋅105.OutputFor each testcase, if it's possible to make a=[1]𝑎=[1], print YES. Otherwise, print NO. You can output the answer in any case (upper or lower). For example, the strings yEs, yes, Yes, and YES will be recognized as positive responses.ExampleinputCopy5101120191000000019000011000outputCopyNoYesNoYesNoNoteIn the fourth testcase of the example, initially a=[1,0,0,0,0,0,0,0,1]𝑎=[1,0,0,0,0,0,0,0,1]. A valid sequence of operations is:Select l=2,r=8𝑙=2,𝑟=8 and apply the operation. a𝑎 becomes [1,0,1][1,0,1].Select l=1,r=3𝑙=1,𝑟=3 and apply the operation. a𝑎 becomes [1][1].