## Play With Mathematical Sequences and Series

Mathematical sequences and series are some of the most useful tools in mathematics. They can be used to solve problems in almost any field. Playing with numbers can be both fun and challenging. These sequences will help you play and learn at the same time! The concepts behind mathematical sequences and series are often presented as a story.

For example, the Fibonacci sequence is often presented as the story of two medieval Italian mathematicians named Fibonacci. These two mathematicians rented land from farmer Giacomo. They would return to the farm every year to collect their money and rent for the following year. The first man would collect his money, plant it in the garden, then go back to town to collect more money from the landlord. After he returned with this cash, he’d dig up some seeds and plant them in the garden where he had stored his savings from last year’s planting.

This process was repeated again and again until one day when they met on opposite sides of a feud that had broken out between their families several years ago. Now, they were forced to live hundreds of miles apart because of this disagreement, which we won’t delve into here.

Here are some of the most famous sequences and series in mathematics. Numbers grow more interesting with a little love. That’s why we picked out some of the best-known sequences and series from the On-Line Encyclopedia of Integer Sequences. You can’t truly appreciate them unless you play with them. So read on to learn how you can play with numbers.

**Recamán’s sequence**, The sequence satisfies [1] a(n) >= 0, [2] |a(n)-a(n-1)| = n, and tries to avoid repeats by greedy choice of a(n) = a(n-1) -+ n. 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155…

**The Busy Beaver problem**, The function Sigma(n) = Sigma(n, 2) (A028444) denotes the maximal number of tape marks (1’s) that a Turing Machine with n internal states (plus the Halt state), 2 symbols, and a two-way infinite tape can write on an initially blank tape (all 0’s) and then halt. 1, 6, 21, 107, 47176870, 8690333381690951…

**The Catalan numbers**, also called Segner numbers. A very large number of combinatorial interpretations are known. The solution to Schröder’s first problem: number of ways to insert n pairs of parentheses in a word of n+1 letters. E.g., for n=2 there are 2 ways: ((ab)c) or (a(bc)); for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ((a(bc))d), (a((bc)d)), (a(b(cd))). 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304…

**The prime numbers**, number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p. A natural number is prime if and only if it has exactly two (positive) divisors. A prime has exactly one proper positive divisor, 1. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271…

**The Mersenne primes**, Equivalently, integers k such that 2^k – 1 is prime. It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K. Length of prime repunits in base 2. 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161…

**The Fibonacci numbers**, in keeping with historical accounts, the generalised Fibonacci sequence a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, … can also be described as the Gopala-Hemachandra numbers H(n) = H(n-1) + H(n-2), with F(n) = H(n) for a = b = 1, and Lucas sequence L(n) = H(n) for a = 2, b = 1. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155…