Shadow sequences of integers, from Fibonacci to Markov and back

Casting light on shadow Somos sequences

“…The solutions of the latter equation are the shadow Somos-4 sequences, in the sense of [21]. However, the equation (3.1) is just the linearization of the Somos-4 recurrence (1.8), and we have the explicit solution of this, which (for fixed coefficients α (0) , β (0) ) depends on four arbitrary parameters; these can be taken to be A (0) , B (0) , z 0 , and J (0) , where the last is the value of the first integral (2.9).…”

“…As a consequence, for the recurrences (1.4) and (1.5), if the initial values are given by x 0 = x 1 = x 2 = x 3 = 1 and any four integers y 0 , y 1 , y 2 , y 3 , then the whole sequence (y n ) consists of integers. More recently, Ovsienko has considered several other examples of nonlinear recurrence relations or birational transformations where each variable x is replaced by a dual number X = x + yε, referring to the corresponding sequence of y values as the shadow sequence [21]. For instance, the Cassini relation for the Fibonacci sequence produces the convolution of the sequence with itself as a shadow, while certain shadow sequences of the Markov numbers appear to be new.…”

“…the source term on the right-hand side) are given in terms of x n . In the special case that α (1) = β (1) = 0 and the right-hand side of (1.9) vanishes, the resulting homogeneous equation is just the linearization of (1.8), corresponding to shadow sequences in the sense of [21].…”

Casting light on shadow Somos sequences

“…The even equation (1.8) is just the ordinary Somos-4 recurrence for x n , with coefficients α (0) , β (0) , while the odd equation (1.9) is an inhomogeneous linear equation for y n , where the coefficients and the inhomogeneity (that is, the source term on the right-hand side) are given in terms of x n . In the special case that α (1) = β (1) = 0 and the right-hand side of (1.9) vanishes, the resulting homogeneous equation is just the linearization of (1.8), corresponding to shadow sequences in the sense of [21].…”