Aronson’s Sequence

Given an integer n, generate the first n terms of the Aronson’s sequence.

Aronson’s sequence is an infinite sequence of integers obtained from the index of T (or t) in the sentence:

“T is the first, fourth, eleventh, sixteenth, … letter in this sentence.”

  • The first occurrence of T in the sentence is at index 1 (1-based indexing) and the number mentioned first is first i.e. 1
  • Similarly, the second occurrence of t in the sentence is at index 4 and the number mentioned second is fourth i.e. 4
  • Similarly, the third occurrence of t in the sentence is at index 11 and the number mentioned third is eleventh i.e. 11
  • Likewise, The series continues as 1, 4, 11, 16, …

Examples:

Input: n = 3
Output: 1, 4, 11

Input: n = 6
Output: 1, 4, 11, 16, 24, 29



Approach: A simple idea is to store the string “T is the” to get the first two terms of the sequence. For each of these terms, convert it to words in the ordinal form and append to the string and calculate the value of the next higher terms. Repeat this process for each of the subsequent higher terms generated for n-2 times to generate the first n terms of the Aronson’s sequence.
For converting a number to words refer here.

Below is the implementation of the above approach:

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// C++ program to generate the
// first n terms of Aronson's sequence
  
#include <bits/stdc++.h>
using namespace std;
  
// Returns the given number in words
string convert_to_words(string num)
{
  
    // Get number of digits in given number
    int len = num.length();
  
    // Base cases
    if (len == 0 || len > 4) {
        return "";
    }
    /*
      The following arrays contain
      one digit(both cardinal and ordinal forms),
      two digit(<20, ordinal forms) numbers, 
      and multiples(ordinal forms) and powers of 10.
      
     */
    string single_digits_temp[]
        = { "", "one", "two", "three", "four"
              , "five", "six", "seven", "eight", "nine" };
    string single_digits[]
        = { "", "first", "second", "third", "fourth"
              , "fifth", "sixth", "seventh", "eighth", "ninth" };
  
    string two_digits[]
        = { "", "tenth", "eleventh", "twelfth", "thirteenth"
              , "fourteenth", "fifteenth", "sixteenth"
              , "seventeenth", "eighteenth", "nineteenth" };
  
    string tens_multiple[]
        = { "", "tenth", "twentieth", "thirtieth", "fortieth"
              , "fiftieth", "sixtieth", "seventieth"
              , "eightieth", "ninetieth" };
  
    string tens_power[] = { "hundred", "thousand" };
    string word = "";
  
    // If single digit number
    if (len == 1) {
        word += single_digits[num[0] - '0'];
        return word;
    }
    int i = 0, ctr = 0;
    string s = " ";
    while (i < len) {
  
        if (len >= 3) {
            if (num[i] != '0') {
                word
                    += single_digits_temp[num[i] - '0']
                       + " ";
  
                // here len can be 3 or 4
                word += tens_power[len - 3] + " ";
                ctr++;
            }
            len--;
            num.erase(0, 1);
        }
  
        // last two digits
        else {
            if (ctr != 0) {
                s = " and ";
                word.erase(word.length() - 1);
            }
  
            // Handle all powers of 10
            if (num[i + 1] == '0')
                if (num[i] == '0')
                    word = word + "th";
                else
                    word += s + tens_multiple[num[i] - '0'];
  
            // Handle two digit numbers < 20
            else if (num[i] == '1')
                word += s + two_digits[num[i + 1] - '0' + 1];
  
            else {
                if (num[i] != '0')
                    word
                        += s + tens_multiple[num[i] - '0']
                               .substr(0, tens_multiple[num[i] - '0']
                                  .length() - 4) + "y ";
                else
                    word += s;
                word += single_digits[num[i + 1] - '0'];
            }
            i += 2;
        }
        if (i == len) {
            if (word[0] == ' ')
                word.erase(0, 1);
        }
    }
    return word;
}
  
// Function to print the first n terms
// of Aronson's sequence
void Aronsons_sequence(int n)
{
    string str = "T is the ";
    int ind = 0;
    for (int i = 0; i < str.length(); i++) {
  
        // check if character
        // is alphabet or not
        if (isalpha(str[i])) {
            ind += 1;
        }
        if (str[i] == 't' or str[i] == 'T') {
            n -= 1;
  
            // convert number to words
            // in ordinal format and append
            str += convert_to_words(to_string(ind)) + ", ";
            cout << ind << ", ";
        }
        if (n == 0)
  
            break;
    }
}
  
// Driver code
int main(void)
{
    int n = 6;
    Aronsons_sequence(n);
    return 0;
}

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Output:

1, 4, 11, 16, 24, 29,


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