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.
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(n1), 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.
 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)
 nonmonotonic  matches non monotonic sequences (1, 3, 2, 1)
 monotonic  matches monotonic sequences (1, 2, 2, 4)
 strictlymonotonic  matches strictly monotonic sequences (1, 2, 3, 4)
 periodic  matches periodic sequences (1,2,3,1,2,3)
 nonperiodic  matches non periodic sequences (1, 2, 3, 2, 3)
 periodictail  matches periodic tails (...1,2,1,2)
 nonperiodictail  matches non periodic tails (...1,2,3)
 irregular  matches irregular sequences (1, 4, 2, 9, 1)
 oeis  matches any sequence with an oeis entry
 linear  matches linear sequences (2,4,6)
 nonlinear  matches nonlinear sequences (2,4,8)
 lineartail  matches linear sequences (...7,8,9)
 nonlineartail  matches nonlinear sequences (...5,8,9)
 convergent  matches linear sequences (...7,8,9)
 divergent  matches nonlinear sequences (...5,8,9)
 periodicn  matches sequences with a period of n
 A000041  matches sequence with the exact OEIS id
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.
 fnintegerconstant  matches functions that use an integer constant (1,2,3...)
 fndecimalconstant  matches functions that use a decimal constant (π,e,ϕ...)
 fnvariable  matches functions that use a variable (n)
 fnarithmetic  matches arithmetic functions (+,,*,/)
 fnbitwise  matches bitwise functions (xor,and,or,...)
 fnrecursive  matches recursive functions (a(n1)...)
 fnspecial  matches special functions (abs, floor...)
 fndivisibility  matches divisibility functions (gcd, %...)
 fnpower  matches power functions (pow, exp, ln...)
 fntrigonometric  matches trigonometric functions (sin, cos...)
 fncombinatoric  matches combinatoric functions (factorial, binomial...)
 fnprime  matches prime functions (p(n), gcd, omega...)
 fnnonintegerconstant  negates the fnintegerconstant tag
 fnnondecimalconstant  negates the fndecimalconstant tag
 fnnonvariable  negates the fnvariable tag
 fnnonarithmetic  negates the fnarithmetic tag
 fnnonbitwise  negates the fnbitwise tag
 fnnonrecursive  negates the fnrecursive tag
 fnnonspecial  negates the fnspecial tag
 fnnondivisibility  negates the fndivisibility tag
 fnnonpower  negates the fnpower tag
 fnnontrigonometric  negates the fntrigonometric tag
 fnnoncombinatoric  negates the fncombinatoric tag
 fnnonprime  negates the fnprime tag
Example
Example Sequences
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
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(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.
Updates
 20181013  Added convergent tag. Improved generation. Added copy
pastable list of sequences. Bug fixes.

20181005  More ops and sequences!

20180928  Big update

20180830  More optimizations to speed up matching. Tag bug fixes.

20180824  Bunch of optimizations to speed up query transform. Bug fixes related to sequence generation.

20180823  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.

20180817  Added products ∏ and fixed some bugs.

20180816  Added summation ∑ and difference Δ
(sub step operations). Restricted max terms in decimal sequences
to 25. Improved sequence generation.

20180713  Stack machines can be executed in sub steps which allows for new functions:
characteristic functions and complement
functions. Also added composite numbers.

20180706  Fixed problem with links.

20180705  Sub sequence matching improved, index title for faster lookup. Bug fixes.

20180702  Added mintermsn to filter out short sequences.
Added paging of search results. Added stern and
abelian group count. Added square operator ².

20180630  Binomial and lookup function bug fixes, fixed approximation searches ~ bug.

20180628  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.

20180621  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.

20180615  Improved search. Using function classes to classify programs (formulas). OEIS tagger fixes.
Ln is now log and a bunch of bug fixes.

20180613  Added lcm(n)=least common multiple. Improved storage model,
paving way for more intelligent generation.

20180608  Improved the expression filter. Better sorting of operators (n*2=>2*n etc).

20180606  Fixed bug that didn't display the decimal expansions.

20180605  Rewrote OEIS matching, better decimal expansion matching and improved sequence generation.

20180601  Added more seeds for recursion. Added
ζ(n)=Riemann Zeta and
zetazero(n)=non trivial zeros of Zeta

20180531  Improved tag descriptions. Discarding periodic sequences that uses lookup functions. Memory
optimizations. Fixed OEIS matching bug.

20180526  Added search tags for function types.

20180525  Some optimizations for searching on tags. Added information on how to search.

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.

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.

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

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.

20170818  Added factorial, sin, cos and tan.

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

20170724  Added absolute value, abs.

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

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.

20170717  Added log and negation() operations. Hover
OEIS links to see description. Decimal expansion
recognition. Many improvements behind the scenes.

20170624  Stack machines can execute with arbitrary precision

20170620  Added the modulo operator %

20170620  Added the recursive function t=a(n3) (generates tribonacci numbers
and more)
 20170617  Added binomial coefficient which will allow Catalan numbers and
similar without too deep searches (thanks @renner96)