Abstract:
We study the statistics of column-convex lattice animals resulting from the stacking of squares on a single or double staircase. We obtain exact expressions for the number of animals with a given length and area, their mean length and their mean height. These objects are closely related to Fibonacci numbers. On a single staircase, the total number of animals with area k is given by the Fibonacci number F_k.

Abstract:
We study the statistics of column-convex lattice animals generated by the stacking of squares on a staircase with step height p. We calculate the number of animals with area k living on l stairs. The total number of animals with area k is given by the generalized Fibonacci number F_p(k). Exact results for the mean length and mean height of animals with area k are also obtained and we examine their asymptotic behaviour.

Abstract:
In the recent study, the new discovery of squaring number was performed. This study is an extension of the study in performing square of numbers be it positive integers or negative integers.

Abstract:
For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e $\lambda=\lambda_1\ge...\ge\lambda_k>0$, and $\lambda_1=k,\lambda_k=1$, then applying the algorithm $\ell$ times gives rise to a set whose cardinality is either the Catalan number $c_{\ell-k+1}$ (the self dual case) or twice the Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two--parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in $2\ell+m$ variables.

Abstract:
The new discovery of squaring number can be use in getting a square of any number be it positive integers or negative integers. Journal of Applied Sciences and Environmental Management Vol. 9(3) 2005: 12-13

Abstract:
The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic factors) $\softO(\sqrt{n})$. Here, the algorithm with time complexity $\softO(n^{2/5})$ and with memory complexity $\softO(n^{1/5})$ is presented. Additionally, a parallel version is shown, which achieves full scalability. As of now the highest computed value was for $n=10^{17}$. Using our implementation we were able to calculate the value for $n=10^{36}$ on a cluster.

Abstract:
From Sturmian and Christoffel words we derive a strictly increasing function $\Delta:[0,\infty)\to\mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of $\Delta$ distinguishes some irrationality measures of real numbers.

Abstract:
Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral.

Abstract:
A word $sigma=sigma_1cdotssigma_n$ over the alphabet$[k]={1,2,ldots,k}$ is said to be a {em staircase} if there are no two adjacent letters with differencegreater than 1.A word $sigma$ is said to be {em staircase-cyclic} if it is a staircase word and in addition satisfies$|sigma_n-sigma_1|le 1$. We find the explicit generating functions for the number of staircase wordsand staircase-cyclic words in $[k]^n$, in terms of {it Chebyshev polynomials of the second kind}.Additionally, we find explicit formul{ae} for the numbers themselves, as trigonometric sums. Theselead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclicwords that are not equivalent up to rotation.