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
OnLine Encyclopedia of Integer Sequences database but with the intention to be
entirely machine generated.
Helping out
If you have a sequence that is not in the database but should be because it can be machine generated.
I will never add a single sequence, but if you have found an underlying generative form that interesting I will try to
add it.
This will capture multiple sequences on the same form. E.g. when the recursive form it automatically resulted
in large amounts of new sequences.
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
Examples
Here are a few examples on different generated sequences.
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(n1),
a(0)=0
.
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(n1), n≥1
.
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(n1)+1)/a(n1), a(0)=2
converges into the golden ratio, ~1.618033988.
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(n1)+(a(n1)a(n2))/n, a(0)=1,
a(1)=2
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(n1)/(11/p(n)), a(0)=2
.
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.
(na(n1))/(a(n2)+a(n2)), 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. The simplest stack machine per function class is chosen. I'm not a mathematician so I have kind of made up
a definition for each function type (constant, n, prime, recursive and primerecursive). What more types could I add?
Future
I don't know where to take this project, here are some ideas.
 Add more operators
 Displayed infix in reduced normal form
 Better search options
 More sequence analysis such as convergence tests
Disclaimer
Most of the code was written on my spare time over with a baby in one arm and my laptop in the
other.
There are at least a few bugs in there. Also I'm aware that some sequence properties may be false. They are derived
from a limited sequence. E.g. it will mark a sequence as monotonic if it appears to be in the first terms.
You should think of it as 'so far monotonic'.
Updates
 20170617  Added binomial coefficient which will allow Catalan numbers and
similar without too deep searches (thanks @renner96)

20170620  Added the recursive function t=a(n3) (generates tribonacci numbers
and more)

20170620  Added the modulo operator %

20170624  Stack machines can execute with arbitrary precision

20170717  Added ln and neg operations. Hover
OEIS links to see description. Decimal expansion
recognition. Many improvements behind the scenes.

20170721  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.

20170723  More robust OEIS matching. Added exp and gcd operations.

20170724  Added absolute value, abs.

20170817  Added sqrt, ceil and round. Fixed high
precision generation bugs that resulted in CPU hog.

20170818  Added factorial, sin, cos and tan.

20170916  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.

20170920  Added Λ(n)=Von Mangoldt's function and Pi
(3.14...).
Also marked seemingly periodic sequences.

20180516  Cross referenced with all new sequences from OEIS.
Fixed bug in OEIS sequence matcher.
Added periodic/monotonic stats the statistics page.
Added periodicn to sequence tags.

20180520  Added P(n)=partition numbers. Added preliminary tag search
integer,
decimal,
monotonic,
strictlymonotonic,
periodicn,
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 1199.
Operators & functions
Sequences are generated from combinations the following operators and functions
Constant
1
2
3
4
5
6
7
8
9
10
Pi (3.141...)
Variable
n
Arithmetic
+
*

/

%
Power
^
ln
exp
sqrt
Prime
p(n)=nth prime
τ(n)=number of divisors of n
ϕ(n)=Euler's totient function
μ(n)=Möbius function
Ω(n)=max factorization terms
λ(n)=Liouville's function
Λ(n)=Von Mangoldt's function
gcd(a,b)=greatest common divisor
Recursive
a(n1)
a(n2)
a(n3)
Trigonometric
sin
cos
tan
Combinatoric
!
P(n)=Partition numbers
C(n,k)=binomial coefficient
Special
floor
abs
ceil
round