Calculus How To

Integer Sequence: Definition & Examples

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A sequence is an ordered list of numbers; An integer sequence is an ordered list of integers (whole numbers plus negatives and zero).

There are hundreds of thousands of integer sequences [1] and hundreds of ways to create them. These range from the simple (adding or multiplying by a constant) to the more complex involving transformations, complementary functions or convolutions [2].

Types of Integer Sequence

Three of the most famous:

  • Natural numbers: The counting numbers (1, 2, 3, …) is the most commonly used integer sequence.
  • Prime numbers: whole numbers (numbers that aren’t fractions) greater than 1 that are divisible only by itself and one. { 2, 3, 5, 7, 11, 13}.
  • Fibonacci numbers: Every term is the sum of the two before it: (0, 1, 1, 2, 3, 5, 8, 13,…).

And three oddball ones:

universal sequence

Could a universal sequence predictor create a universe?

  • The universal sequence is a sequence that can (theoretically) create every sequence in the universe. The computing power to run such a sequence doesn’t exist (and never will).
  • Recamán’s Sequence: A fun sequence that seems to have no purpose other than to confuse with its mysterious outputs.
  • Constant Sequence: A sequence that has all of the same numbers. For example, {1, 1, 1, 1, 1, …).

Uses of Integer Sequence

Integer sequence appear in just about every branch of mathematics and science, including [3]:

  • Chemistry (e.g. atom cluster sizes).
  • Computer science (e.g. number of steps required to sort x items).
  • Enumeration problems (e.g. combinatorics, graph theory, lattices).
  • Number theory (for example, a list of solutions to x2 + y2 + z2?).
  • Physics (e.g. paths on lattices).

Who Invented the First Integer Sequence?

The earliest mathematical artifact known to man is the Lebombo bone (c. 33, 000) B.C., which has 29 tally marks; It is the oldest sequence listed in the OEIS. The British Museum has an old Babylonian clay cuneiform tablet, containing a table of squares and cubes [4]. But our knowledge is limited to those artifacts that are in existence: most are lost to time.

The real question is, who created the integers? Because once they were created, the sequence would have fallen into place. Perhaps Stephen Hawking was right when he said that “God Created The Integers” [5] (or perhaps it was his was of saying “We’ll never know”).


[1] The Online Encyclopedia of Integer Sequences (OEIS).
[2] Khovanova, T. (2007). How to Create a New Integer Sequence. Retrieved April 10, 2021 from:
[3] Sloane, N. My Favorite Integer Sequences. Retrieved April 10, 2021 from:
[4]. The British Museum. Tablet: 92698.
[5] Hawking, S. (2005). God Created the Integers: The Mathematical Breakthroughs that Changed History. Running Press.

Stephanie Glen. "Integer Sequence: Definition & Examples" From Calculus for the rest of us!

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