Survey of Cubic Fibonacci Identities When Cuboids Carry Weight

The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter of mathematical curiosity, but also motivated by a quest for 3D Fibonacci spirals. As we were not able to find any survey on the particular topic of cubic Fibonacci identities we decided to try to fill this void. We started by surveying many Fibonacci identities and recording cubic ones. Obviously, tracing all Fibonacci identities (for identifying a handful) is a daunting task. Checking several hundred we have realized that it is not always clear who the author is. The reason is that in many cases an identity was stated in one article (sometimes without a proof, e.g., as an open problem, or a conjecture) while later being proven in another article, or effectively rediscovered independently by other authors. However, we have done our best to present the identities chronologically. We have supplied our own proof for one identity, having tried, but failed, to find a published proof. For all the other identities, we either proved them on a computer, or else verified by hand their original published proofs. Somehow unexpectedly, our investigations have revealed only a rather small number of cubic Fibonacci identities, representing a tiny fraction of all published Fibonacci identities (most of which are linear or quadratic). Finally, out of these, only a handful of cubic Fibonacci identities are homogeneous

Checking whether a word is Hamming-isometric in linear time

A finite word $f$ is Hamming-isometric if for any two word $u$ and $v$ of same length avoiding $f$, $u$ can be transformed into $v$ by changing one by one all the letters on which $u$ differs from $v$, in such a way that all of the new words obtained in this process also avoid~$f$. Words which are not Hamming-isometric have been characterized as words having a border with two mismatches. We derive from this characterization a linear-time algorithm to check whether a word is Hamming-isometric. It is based on pattern matching algorithms with $k$ mismatches. Lee-isometric words over a four-letter alphabet have been characterized as words having a border with two Lee-errors. We derive from this characterization a linear-time algorithm to check whether a word over an alphabet of size four is Lee-isometric.Comment: A second algorithm for checking whether a word is Hamming-isometric is added using the result given in reference [5

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue--Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue--Morse sequence.Comment: 23 pages, 5 figures, 4 table

Moments of Catalan Triangle Numbers

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section