We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
Abstract
In the paper, there has been constructed such a non-classical Bittner operational calculus model, in which the derivative is understood as a central difference Dn{x(k)}:= {x(k+n)–x(k-n)}. The discussed model has been generalized by considering the operation Dn,b{x(k)}:= {x(k+n)–bx(k-n)}, where b∈𝕔\{0}. In the D1-difference model exponential-trigonometric and hyperbolic Fibonacci sequences have been introduced.
Some results on circulant and skew circulant type matrices with k-Fibonacci sequences
