OEIS/Holonomic Project

This project tries to define as many existing OEIS sequences as possible by holonomic recurrences.

Whereas the usual linear recurrence has constant coefficients, a holonomic recurrence has coefficients that are polynomials in n (the sequence index). Holonomic recurrences can be derived from linear differential equations, and are applicable to a wide class of series - see Wikipedia: Holonomic function.

Hierarchy of recurrences

There is a list of sequence types which are included in this project, with increasing complexity and computational effort:

• constant
• finite
• periodic
• recurrent with constant coefficients = defined by a rational generating function (fraction of tow polynomials in x)
• holonomic recurrent

Embedding in jOEIS

jOEIS is a project which aims to implement all OEIS sequences in Java. Currently (Feb. 2021) there are more than 104,000 sequences implemented. They are compiled in a single jar file, and any sequence can be called repeatedly in a uniform way to produce the next terms.

In jOEIS there is a series of basic Java classes which generate sequences of the holonomic type. They can all be found under https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis:

BlockMultAddSequence
ContinuedFractionOfSqrtSequence
CoordinationSequence
CoxeterSequence
FiniteSequence
GeneratingFunctionSequence
HolonomicRecurrence
LatticeCoordinationSequence
LinearRecurrence
PaddingSequence
PeriodicSequence
PrependSequence