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Building Lists from Functions

Array[f,n]generate a length n list of the form {f[1], f[2], ...}
Array[f,{n1,n2,...}]generate an n1×n2×... nested list, each of whose entries consists of f applied to its indices
NestList[f,x,n]generate a list of the form {x, f[x], f[f[x]], ...}, where f is nested up to n deep
FoldList[f,x,{a,b,...}]generate a list of the form {x, f[x, a], f[f[x, a], b], ...}
ComposeList[{f1,f2,...},x]generate a list of the form {x, f1[x], f2[f1[x]], ...}

Making lists from functions.

This makes a list of 5 elements, each of the form p[i].
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Here is another way to produce the same list.
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This produces a list whose elements are i+i2.
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This generates a 2×3 matrix whose entries are m[i, j].
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This generates a 3×3 matrix whose elements are the squares of the sums of their indices.
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NestList and FoldList were discussed in "Applying Functions Repeatedly". Particularly by using them with pure functions, you can construct some very elegant and efficient Mathematica programs.
This gives a list of results obtained by successively differentiating xn with respect to x.
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