A tiling involution for the Sury’s identity

We study integer sequences defined by the recurrence Un+2 = p Un+1 + Un and the initial values U0 = a, U1 = 1, for n ≥ 0. We find families of identities of these sequences, some of which Sury’s identities are a special case. We prove these identities by using a combinatorial interpretation by means of tiling. In particular, we present a tiling involution of the alternating sign dual of the first Sury’s identity.