# SequenceFoldList

SequenceFoldList[f,{x1,,xn},{a1,a2,}]

gives {x1,,xn,f[x1,,xn,a1],f[x2,,xn,f[x1,,xn,a1],a2],}.

SequenceFoldList[f,{x1,,xn},{a1,a2,},k]

applies f to k arguments at each step, with the first n coming from the xi or previous results, and the last k-n coming from the ai.

# Details • In SequenceFoldList[f,{x1,,xn},{a1,a2,}], the function f is assumed to take n+1 arguments.
• If xlist and alist are lists of length n and m respectively, then SequenceFoldList[f,xlist,alist] returns a list of length n+m. Any or both of n and m can be zero.
• SequenceFoldList[f,{x},{a,b,}] is equivalent to FoldList[f,x,{a,b,}].
• SequenceFoldList[f,{x1,,xn},{a1,a2,}] is equivalent to SequenceFoldList[f,{x1,,xn},{a1,a2,},n+1].
• If xlist and alist are lists of length n and m respectively, then SequenceFoldList[f,xlist,alist,k] returns a list of length Max[2n+m-k+1,n]. Any of n, m, k can be zero, as long as kn.

# Examples

open allclose all

## Scope(2)

Take one element of the third argument at each step:

Take several elements of the third argument at each step:

## Applications(2)

Fibonacci sequence:

Show convergence to the exact solution when increasing the order of extrapolation:

## Properties & Relations(10)

SequenceFold[f,xlist,alist] returns the last element of SequenceFoldList[f,xlist,alist]:

Each element of the result of SequenceFoldList[f,xlist,alist] is constructed from the previous one and an element of alist:

Each element of the result of SequenceFoldList[f,xlist,alist,k] is constructed from the previous one and several elements of alist:

SequenceFold gives the last element of the result of SequenceFoldList:

Starting with a seed sequence of length 1 makes SequenceFoldList equivalent to FoldList:

An empty seed sequence makes SequenceFoldList equivalent to Map:

If the list in the third argument is empty, then the seed sequence is returned:

This is equivalent to applying f to a partitioned list:

If f has only one argument, then SequenceFoldList is equivalent to NestList:

The number of arguments of the function cannot be smaller than the length of the seed sequence: If they are equal, then the actual elements of the list in the third argument are not used: