Needs["IntegerSequences`"]

values of a sequence

ruler=RegularSequence[{1,0},{{{0,0},{-1,1}},{{0,1},{-1,2}}},{0,1}];

Table[ruler[n],{n,0,15}]

closure properties

SternBrocot/@Range[0,15]

Table[((SternBrocot[n])/(SternBrocot[n+1])),{n,0,15}]

RegularSequenceReduce[2 SternBrocot[n]+ThueMorse[n]ruler[n],n]

RegularSequenceMatrixForm[%]

guessing

number of 1s in binary

terms=Table[DigitCount[n,2,1],{n,0,50}]

ListLinePlot[terms]

FindRegularSequenceFunction[terms,2]//RegularSequenceMatrixForm

Or guess a recurrence:

FindRegularSequenceRecurrence[terms,2,s[n]]//Column

multidimensional sequences

ArrayPlot[array=Table[IntegerExponent[Binomial[n,m],2],{n,0,100},{m,0,100}]]

FindRegularSequenceFunction[array/. ∞->infinity,2]//RegularSequenceMatrixForm

extremal words

Lexicographically least sequence avoiding squares:

Table[ruler[n],{n,0,15}]

0,1,0

Lexicographically least sequence avoiding (3/2)-powers:

ListLinePlot[{0,0,1,1,0,2,1,0,0,1,1,2,0,0,1,1,0,3,1,0,0,1,1,3,0,0,1,1,0,2,1,0,0,1,1,4,0,0,1,1,0,3,1,0,0,1,1,2,0,0,1,1,0,2,1,0,0,1,1,3,0,0,1,1,0,3,1,0,0,1},PlotRange->All]

FindRegularSequenceFunction[{0,0,1,1,0,2,1,0,0,1,1,2,0,0,1,1,0,3,1,0,0,1,1,3,0,0,1,1,0,2,1,0,0,1,1,4,0,0,1,1,0,3,1,0,0,1,1,2,0,0,1,1,0,2,1,0,0,1,1,3,0,0,1,1,0,3,1,0,0,1},6]

RegularSequenceMatrixForm[%]

Theorem (Rowland–Shallit 2012).
The lexicographically least (3/2)-power-free sequence is 6-regular.

Theorems (Pudwell–Rowland 2018).
The lexicographically least (5/3)-power-free sequence is 7-regular.
The lexicographically least (8/5)-power-free sequence is 733-regular.
The lexicographically least (7/4)-power-free sequence is 50847-regular.
etc. etc.

computing automata for sequences modulo (p^α)

Catalan numbers modulo 3

Table[Mod[CatalanNumber[n],3],{n,0,15}]

AutomaticSequenceReduce[Mod[CatalanNumber[n],3],n]

AutomatonGraph[AutomaticSequenceAutomaton[%]]

VertexCount[%]

diagonal of a rational power series

AutomaticSequenceReduce[Mod[DiagonalSequence[((1-x)/(1-(((1+x))^2) y)),{x,y}][n],4],n]

AutomatonGraph[AutomaticSequenceAutomaton[%]]

Motzkin numbers modulo 8

Table[MotzkinNumber[n],{n,0,10}]

What values does the sequence of Motzkin numbers take modulo 8?

AutomatonGraph[automaton=AutomaticSequenceAutomaton[AutomaticSequenceReduce[Mod[MotzkinNumber[n],8],n]],VertexSize->.5]

Sort[AutomatonOutputAlphabet[automaton]]

Motzkin numbers modulo (13^2)

Theorem (Rowland–Yassawi 2015).
No Motzkin number is divisible by (13^2).

Apéry numbers modulo 16

Table[AperyNumber[n],{n,0,10}]

Gessel (1982) proved that the Apéry numbers are periodic modulo 8:

Table[Mod[AperyNumber[n],8],{n,0,15}]

He asked whether they are periodic modulo 16.

Table[Mod[AperyNumber[n],16],{n,0,15}]

AutomatonGraph[automaton=AutomaticSequenceAutomaton[AutomaticSequenceReduce[Mod[AperyNumber[n],16],n,Method->"Diagonal",Monitor->True]],VertexSize->.4]

Theorem (Rowland–Yassawi 2015).
The Apéry numbers are not eventually periodic modulo 16.