[seqfan] 3 nonlinear recurrences

[Georgi Guninski]

[1]
A059480 A recurrence equation.
a(n) = a(n - 1) + (n + 1)*a(n - 2)

a[i+1]=( a[i-2]*a[i-1]-a[i-1]^2+a[i-2]*a[i]+a[i-1]*a[i])/a[i-2]
(i think i proved this by substituting the definition and simplifying to
0)

[2]
A135686 Two sequence type recursion of a generalized Stirling number
type using the Fibonacci sequence

a[n]=a[n-1]+fibonacci[n]*a[n-2]

w0,w1,w2,w3,w4,w5,w6,w7= -1, 1, -1, -1, -1, 1, 1, 1
a[i+2] =(w0*a[i-1]^2*a[i]  -
(w1*a[i-2]*a[i]^2+w2*a[i-1]*a[i]^2+w3*a[i-2]*a[i-1]*a[i+1]+w4*a[i-2]*a[i]*a[i+1]))/(w5*a[i-2]*a[i-1])

[3]
a[0]=0,a[1]=1
a[n]=a[n-1]+2^n * a[n-2]

0, 1, 1, 9, 25, 313, 1913, 41977, 531705, 22023929, 566489849

a[i+1]=(-2*a[i-1]^2+a[i-2]*a[i]+2*a[i-1]*a[i])/a[i-2]