About

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.

Contact

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

Search

Searching is simple, just enter a few sequential terms and hit enter.

Example

Search by formula

To find all sequences matching a formula, just enter it in the search bar. You can combine it with sequence terms and tags. For now, the entered formula has to match with the generated so it can be hard to get it fully right, however specifying parts of the formula is enough.

Example

Matching sequences containing 3,4 that are recursive, a(n-1), and uses partition numbers, P.

Search by sequence tags

You can combine a sequence with multiple tags to narrow down the search. The sequence tags only describe the first 50 terms so you should think of it as 'so far monotonic', 'so far periodic' etc.

Example

Search by function tags

You can filter your searches on function tags. I have grouped functions a bit arbitrary, let me know if you find something weird.

Example

Example Sequences

Here are a few examples on different generated sequences.

Square numbers

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576
This common sequence is generated either directly by n*n or recursively by n+a(n-1), a(0)=0.

Prime gaps

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4
Gaps between consecutive primes, generated by p(n)-p(n-1), n≥1.

Golden Ratio

2, 1.5, 1.6666666667, 1.6, 1.625, 1.6153846154, 1.619047619, 1.6176470588, 1.6181818182, 1.6179775281, 1.6180555556, 1.6180257511, 1.6180371353, 1.6180327869, 1.6180344478, 1.6180338134, 1.6180340557, 1.6180339632, 1.6180339985, 1.618033985, 1.6180339902, 1.6180339882, 1.618033989, 1.6180339887, 1.6180339888
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.

Euler e

1, 2, 2.5, 2.6666666667, 2.7083333333, 2.7166666667, 2.7180555556, 2.7182539683, 2.7182787698, 2.7182815256, 2.7182818011, 2.7182818262, 2.7182818283, 2.7182818284, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285, 2.7182818285
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

Euler Zeta, s=1

1, 2, 3, 3.75, 4.375, 4.8125, 5.2135416667, 5.5393880208, 5.8471317998, 6.1129105179, 6.3312287507, 6.5422697091, 6.7239994232, 6.8920994088, 7.0561970138, 7.209592601, 7.3482386126, 7.4749323818, 7.5995145881, 7.7146587486, 7.8248681593, 7.9335468837, 8.0352590232, 8.1332499869, 8.2256732822, 8.3113573789
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.

Recursive Waves

Finally a sequence I ran into by chance when I was adding charts to the site. I've no idea what it means or if it has any value. But it looks pretty funky. (n-a(n-1))/(a(n-2)+a(n-2)), a(0)=1, a(1)=2

Algorithm

The algorithm generates stack machines that are executed with different input (0≤n<N). The output of each execution form a sequence that is stored. Duplicated sequences are detected, so that different stack machines can generate the same sequence.

Updates