Related papers: Integer Sequences and Output Arrays
In this paper, we discuss about some results on average of Fibonacci and Lucas sequences that may be found in the OEIS code A111035.
The OEIS sequence A051221 consists of nonnegative integers of the form 10^x - y^2. The known values are those less than or equal to 2000 with x <= 7, and it is conjectured that no new values in this range appear for x >= 8. In this paper,…
Tight bounds on the block entropy of patterns of sequences generated by independent and identically distributed (i.i.d.) sources are derived. A pattern of a sequence is a sequence of integer indices with each index representing the order of…
We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are…
This paper introduces a class of extended central factorial numbers generated by a parity-dependent recurrence relation, termed the "flickering operator". We demonstrate that the resulting triangular structure, now indexed as OEIS A395021,…
We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1)^{n-1}A_n=C_n+\sum_{j=1}^{n-1} (-1)^{j} \binom{2n-1}{2j-1} A_j \,C_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$)…
In an automatic search, we found conjectural recurrences for some sequences in the OEIS that were not previously recognized as being D-finite. In some cases, we are able to prove the conjectured recurrence. In some cases, we are not able to…
We introduce the $2$-regular integer sequence A383066 $= (s(n))_{n \geq 1}$, which begins $0, 1, 1, 2, 3, 3, 2, \ldots$. We prove that the number of occurrences of an integer $m \geq 0$ in this sequence is equal to $\tau(m^2+1)$, the number…
Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…
Ascent sequences have received a lot of attention in recent years in connection with (2 + 2)-free posets and other combinatorial objects. Here, we first show bijectively that analogous repetition sequences are counted by the Bell numbers,…
We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…
We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce…
In this paper we determine the parity of some sequences which are related to Catalan numbers. Also we introduce a combinatorical object called, \Catalan tree", and discuss its properties.
In 2003, Benoit Cloitre entered a family of sequences in the OEIS that we call hiccup sequences. We collect the various claims, observations, and proofs of properties of these sequences that have been entered in the OEIS over the years, and…
Action graphs emerged from work of Bergner and Hackney on category actions in the context of Reedy categories. Alvarez, Bergner, and Lopez showed that action graphs could be inductively generated without reference to category actions and…
The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we…
Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the…
A sequence $a_0<a_1<\ldots<a_n$ of nonnegative integers is called a Sidon sequence if the sums of pairs $a_i+a_j$ are all different. In this paper we construct CAT(0) groups and spaces from Sidon sequences. The arithmetic condition of Sidon…
The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting…
We prove that five ways to define entry A086377 in the On-Line Encyclopedia of Integer Sequences do lead to the same integer sequence.