Sequencer – purely algorithmic number sequence identification
Sequencer identifies number sequences. That is, given a list of numbers like
(a(n)) = 1, 2, 4, 8, 16, 32, ...
it finds a formula that generates them, in this case
a(n) = 2^(n-1)
Sequencer employs neither a library of sequences nor a limited set of algorithms to find a closed form. Instead, it generates all formulas up to a certain size and then checks them against the provided numbers.
For verification, the system uses a hybrid approach of a fast numerical checker followed by a symbolic verifier powered by the Symja computer algebra system. Coupled with some tricks and heuristics designed to quickly generate potentially interesting formulas, Sequencer can identify sequences with very complex closed forms in a matter of seconds when run on commodity hardware.
Sequencer is capable of finding closed forms that are beyond any existing system like OEIS, Superseeker and Wolfram Alpha. It is particularly strong where recurrence relations or unusual combinations of functions are involved. For example, none of the services mentioned above can currently make sense of the sequence
(a(n)) = 1, 1, 1, 3, 5, 15, 43, 273, ...
while Sequencer reveals that it satisfies the recurrence relation
a(1) = 1 a(2) = 1 a(3) = 1 a(n) = a(n-2)^2+a(n-1)+a(n-3) for n >= 4
and provides the continuation
2137, 76709, 4643751, 5888916569, 21570312343279, ...
Symbolic input and output
Sequencer is not limited to processing integers but can identify sequences consisting of arbitrary Symja expressions (provided they can be evaluated numerically). For example, invoking the program with the arguments
0 1/2 sqrt(3)/2 1 produces
a(n) = Sin(1/6*Pi*(n-1)) Continuation: 1/2*3^(1/2), 1/2, 0, -1/2, (-1/2)*3^(1/2), ...
Note that parentheses in arguments need to be escaped (
\() when running a program from a shell like bash.
Installation and usage
java -jar sequencer.jar 1 2 3 4 5
Running the program without arguments displays a help text explaining the various command line parameters that can be used to fine-tune how searches are performed.
Sequencer can also be used as a library, for which precompiled JARs (
sequencer-library-X.X.X.jar) are available on the releases page.
Sequencer provides the method
def identifySequence(sequence: Seq[String]): Seq[SequenceIdentification]
that returns objects of type
case class SequenceIdentification(formula: String, continuation: Seq[String])
When instantiating, the class must be passed a
case class Configuration( maximumComplexity: Int, maximumIdentifications: Int, predictionLength: Int, recurrenceDepth: Int, combinatorialFunctions: Boolean, numberTheoreticFunctions: Boolean, transcendentalFunctions: Boolean, parallelSearch: Boolean, numericalTest: Boolean, printProgress: Boolean, outputLaTeX: Boolean )
that controls the behavior of
identifySequence. For more details, see the source code.
Sequencer is written in Scala. To compile Sequencer from source, you need Git, a JDK, the Scala compiler, and sbt. Once all of these are installed and on your
PATH, you are ready to build and run Sequencer:
git clone https://github.com/p-e-w/sequencer.git cd sequencer sbt run
The standalone JAR can be created using
The library JAR can be created using
All generated JARs will be written to
Besides its runtime and compilation environment (Java, Scala and sbt), Sequencer depends on the Symja computer algebra system and the scopt command line parser. The standalone release JARs are built using the excellent sbt-assembly plugin.
Copyright © 2015 Philipp Emanuel Weidmann (firstname.lastname@example.org)
Released under the terms of the GNU General Public License, Version 3