This is an experiment that was created out of curiosity. The purpose is to see how computer generated sequences take form given certain restrictions. It's inspired by the great On-Line Encyclopedia of Integer Sequences database but with the intention to be entirely machine generated.

If you have any feedback or questions, please don't hesitate to contact me. You can contact me at jon AT jonkagstrom DOT com or via twitter

- constant - matches constant sequences (1, 1, 1)
- integer - matches integer sequences (1, 2, 3)
- decimal - matches decimal sequences (1, 2.2, 3)
- positive - matches positive sequences (1, 2, 3)
- negative - matches negative sequences (-1, -2, -3)
- non-monotonic - matches non monotonic sequences (1, 3, 2, 1)
- monotonic - matches monotonic sequences (1, 2, 2, 4)
- strictly-monotonic - matches strictly monotonic sequences (1, 2, 3, 4)
- non-periodic - matches non periodic sequences (1, 2, 3, 2, 3)
- non-periodic-tail - matches non periodic tails (...1,2,3)
- periodic - matches periodic sequences (1,2,3,1,2,3)
- oeis - matches any sequence with an oeis entry
- irregular - matches irregular sequences (1, 4, 2, 9, -1)
- periodic-n - matches sequences with a period of n
- A000041 - matches sequence with the exact OEIS id

- fn-constant - matches functions that use a constant (1,2,3...)
- fn-variable - matches functions that use a variable (n)
- fn-arithmetic - matches arithmetic functions (+,-,*,/)
- fn-recursive - matches recursive functions (a(n-1)...)
- fn-special - matches special functions (abs, floor...)
- fn-divisibility - matches divisibility functions (gcd, %...)
- fn-power - matches power functions (pow, exp, ln...)
- fn-trigonometric - matches trigonometric functions (sin, cos...)
- fn-combinatoric - matches combinatoric functions (factorial, binomial...)
- fn-prime - matches prime functions (p(n), gcd, omega...)
- fn-non-constant - negates the fn-constant tag
- fn-non-variable - negates the fn-variable tag
- fn-non-arithmetic - negates the fn-arithmetic tag
- fn-non-recursive - negates the fn-recursive tag
- fn-non-special - negates the fn-special tag
- fn-non-divisibility - negates the fn-divisibility tag
- fn-non-power - negates the fn-power tag
- fn-non-trigonometric - negates the fn-trigonometric tag
- fn-non-combinatoric - negates the fn-combinatoric tag
- fn-non-prime - negates the fn-prime tag

This common sequence is generated either directly by

`n*n`

or recursively by ```
n+a(n-1),
a(0)=0
```

.
Gaps between consecutive primes, generated by

`p(n)-p(n-1), n≥1`

.
Decimal sequences may converge into interesting constants, the generated program

`(a(n-1)+1)/a(n-1), a(0)=2`

converges into the golden ratio, ~1.618033988.
Here is another famous constant e (2.7182818285...) generated by

```
a(n-1)+(a(n-1)-a(n-2))/n, a(0)=1,
a(1)=2
```

This sequence is probably the reason I built this site. A while back I was playing with the Sieve of Eratosthenes.
By counting how often each term strike out following composite numbers this sequence appeared.
3 strikes out every second, 5 every third, 7 every 3.75, 11 every 4.375 and so on. I had no idea that this was
Euler Zeta with exponent 1 at the time. I asked a friend who is very good at maths to figure out the formula,
and after a while he came back with the answer. Somewhere here I figured that a program should be able to give me
answers, so I don't have to ask my friend all the time :) This formula is closely related to the Riemann hypothesis.
The generated code is

`a(n)=a(n-1)/(1-1/p(n)), a(0)=2`

.
`(n-a(n-1))/(a(n-2)+a(n-2)), a(0)=1, a(1)=2`

- 2018-08-30 - More optimizations to speed up matching. Tag bug fixes.
- 2018-08-24 - Bunch of optimizations to speed up query transform. Bug fixes related to sequence generation.
- 2018-08-23 - If a sequence query give no matches, a bunch of transforms will be applied, e.g. 1, 7, 27, 76, 175, 351, 637, 1072, 1701, 2575, 3751, 5292, 7267, 9751.
- 2018-08-17 - Added products ∏ and fixed some bugs.
- 2018-08-16 - Added summation ∑ and difference Δ (sub step operations). Restricted max terms in decimal sequences to 25. Improved sequence generation.
- 2018-07-13 - Stack machines can be executed in sub steps which allows for new functions: characteristic functions and complement functions. Also added composite numbers.
- 2018-07-06 - Fixed problem with links.
- 2018-07-05 - Sub sequence matching improved, index title for faster lookup. Bug fixes.
- 2018-07-02 - Added min-terms-n to filter out short sequences. Added paging of search results. Added stern and abelian group count. Added square operator ².
- 2018-06-30 - Binomial and lookup function bug fixes, fixed approximation searches ~ bug.
- 2018-06-28 - Putting formulas between '[' and ']' this will help the query parser. For example [log(2 * n)]. Added aliases for operations e.g. π=pi Instead of ignoring errors in the search query, they are displayed.
- 2018-06-21 - Added σ(n)=divisor sum of n, pt(n)=Pascals triangle rows, lpf(n)=least prime factor of n and gpf(n)=greatest prime factor of n. Improved generation. Disabled auto capitalization for mobiles.
- 2018-06-15 - Improved search. Using function classes to classify programs (formulas). OEIS tagger fixes. Ln is now log and a bunch of bug fixes.
- 2018-06-13 - Added lcm(n)=least common multiple. Improved storage model, paving way for more intelligent generation.
- 2018-06-08 - Improved the expression filter. Better sorting of operators (n*2=>2*n etc).
- 2018-06-06 - Fixed bug that didn't display the decimal expansions.
- 2018-06-05 - Rewrote OEIS matching, better decimal expansion matching and improved sequence generation.
- 2018-06-01 - Added more seeds for recursion. Added ζ(n)=Riemann Zeta and Z(n)=non trivial zeros of Zeta
- 2018-05-31 - Improved tag descriptions. Discarding periodic sequences that uses lookup functions. Memory optimizations. Fixed OEIS matching bug.
- 2018-05-26 - Added search tags for function types.
- 2018-05-25 - Some optimizations for searching on tags. Added information on how to search.
- 2018-05-20 - Added P(n)=partition numbers. Added preliminary tag search integer, decimal, monotonic, strictly-monotonic, periodic-n, oeis and oeis id. You can combine those in the search bar with the sequence, e.g. 1, 2, 3, 5, 6, ln integer oeis, more info later. Added constants 11-99.
- 2018-05-16 - Cross referenced with all new sequences from OEIS. Fixed bug in OEIS sequence matcher. Added periodic/monotonic stats the statistics page. Added periodic-n to sequence tags.
- 2017-09-20 - Added Λ(n)=Von Mangoldt's function and Pi (3.14...). Also marked seemingly periodic sequences.
- 2017-09-16 - Added prime related functions τ(n)=number of divisors of n, ϕ(n)=Euler's totient function, μ(n)=Möbius function, Ω(n)=max factorization terms and λ(n)=Liouville's function.
- 2017-08-18 - Added factorial, sin, cos and tan.
- 2017-08-17 - Added sqrt, ceil and round. Fixed high precision generation bugs that resulted in CPU hog.
- 2017-07-24 - Added absolute value, abs.
- 2017-07-23 - More robust OEIS matching. Added exp and gcd operations.
- 2017-07-21 - Using 50 terms instead of 25 for indexing. Added floor. Use high precision at top when opening sequence. Added ~ for approximate searches (each term within 5%) e.g. ~17,20,26,33,41,46,55,66. Bunch of bug fixes.
- 2017-07-17 - Added log and negation(-) operations. Hover OEIS links to see description. Decimal expansion recognition. Many improvements behind the scenes.
- 2017-06-24 - Stack machines can execute with arbitrary precision
- 2017-06-20 - Added the modulo operator %
- 2017-06-20 - Added the recursive function t=a(n-3) (generates tribonacci numbers and more)
- 2017-06-17 - Added binomial coefficient which will allow Catalan numbers and similar without too deep searches (thanks @renner96)