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C00002 00002	Palindromic continued fractions
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Palindromic continued fractions

Knuth, vol. 2 defines Qi(x1,...,xi) by

Q[i](x1,...,xi) ← if i=0 then 1
		else if i=1 then x1
		else x1*Q(i-1)(x2,...xn) + Q[i-2](x3,...xn)

Experiment with MACSYMA has led to the conjectures

Q[2k-1](x2,...xk,xk,...,x1)↑2 + 1 =
	= Q[2k](x1,...,xk,xk,...,x1)*(Q[k-1](x2...xk)↑2 + Q[k-2](x2...x[k-1])↑2)

Q[2k](x2...x[k+1],xk...x1)↑2 - 1 =
	= Q[2k+1](x1...x[k+1]...x1)*Q[2k-1](x2...x[k+1]...x2)

Q[2k](x1...xk,xk...x1) = Q[k](x1...xk)↑2 + Q[k-1](x1...x[k-1])↑2

We should also remember the identity

Q[n](x1...xn)*Q[n](x2...x[n+1]) - Q[n+1](x1...x[n+1])*Q[n-1](x2...xn) = (-1)↑n.