Nonlinear recurrence relation

Just for fun: exploring critical points

f[x_, y_] := (1 + y + x^3)/3 nf[a0_, a1_, n_] := NestList[{#[[2]], f[#[[1]], #[[2]]]} &, {a0, a1}, n] g = {#1, #2, f[#1, #2]} &; cp = {a, b} /. Solve[nf[a, b, 1] == {a, b}, {a, b}]; 

A limited exploration:

vis[a0_, a1_, n_] := With[{pt = g @@@ nf[a0, a1, n]}, Show[Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, PlotRange -> {-10, 10}, Mesh -> None, PlotStyle -> Opacity[0.4]], Graphics3D[{Black, PointSize[0.03], Point[{a0, a1, f[a0, a1]}], PointSize[0.01], Red, Point[pt], Line[pt], Green, PointSize[0.03], Point[g @@@ cp]}]] ] 

In my process of learning, I am always encountering many recursive formular. The answer as shown below is my summarization about dealing with these recursive formular in Mathematica's functional paradigm.

For the general formular $$a_{n+1}=func(a_{n-1},a_n)$$ you can use the a general solution

# & @@@NestList[{#2, $func$[#1,#2]} & @@ # &, {$a_1$, $a_2$}, $n$]

where $n$ is the $n$th number

So in your question :$$func(x,y)=\frac{x^3+y+1}{3}$$

# & @@@ NestList[{#2, 1/3 (#1^3 + 1 + #2)} & @@ # &, {1, 2}, 5] 

{1, 2, 4/3, 31/9, 184/81, 32176/2187, 14579713/1594323} 

Other example

Fibonacci

$a_{n+1}=a_{n-1}+a_n$

# & @@@ NestList[{#2, #1+#2} & @@ # &, {1, 1}, 10] 

 {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89} 

Double sequences

$$x_n=\frac{1}{2}\left(x_{n-1}+\sqrt{1-y_{n-1}{}^2}\right) \\ y_n=\frac{1}{2}\left(y_{n-1}+\sqrt{1-x_n{}^2}\right)$$

So you can do like this:

func[x_, y_] := 1/2 (Sqrt[1 - y^2] + x); NestList[ Function[{x, y}, {func[x, y], func[y, func[x, y]]}] @@ # &, {.5, Sqrt[3]/4}, 10] 

{{0.5, Sqrt[3]/4}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}, {0.780556, 0.624467}, {0.780804, 0.624622}, {0.780865, 0.624661}, {0.780881, 0.62467}, {0.780885, 0.624673}, {0.780886, 0.624673}} 

Or

NestList[ {1/2 (Sqrt[1 - #2^2] + #1), 1/2 (#2 + Sqrt[1 - (1/2 (Sqrt[1 - #2^2] + #1))^2])} & @@ # &, {.5, Sqrt[3]/4}, 10] 

Related

• How to deal with recursion formula in Mathematica?

---

Reference

• Sequence generation

One way to approach this is to write the equation recursively:

a[n_] := a[n] = (1 + a[n - 1] + a[n - 2]^3)/3; a[1] = a1; a[0] = a0; 

This leaves the initial conditions in terms of two generic parameters a0 and a1. For example, a[3] gives

1/3 (1 + a1^3 + 1/3 (1 + a0^3 + a1)) 

FullSimplify[a[4]] is:

1/81 (27 + (1 + a0^3 + a1)^3 + 3 (4 + a0^3 + a1 + 3 a1^3)) 

and so on. This gives an expression for a[n] expanded in terms of the initial conditions. If you wish a numerical answer, assign a0 and a1 to specific values.