Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle

Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-αx+1$ for $α=3,4,5,\dots$.Some of the underlying identities suggest an identity involving binomialcoefficients which leads to a triangular array sharing many propertieswith Pascal's triangle.

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