scholarly journals Several explicit and recursive formulas for generalized Motzkin numbers

2020 ◽  
Vol 5 (2) ◽  
pp. 1333-1345
Author(s):  
Feng Qi ◽  
◽  
Bai-Ni Guo ◽  
◽  
Keyword(s):  
Motzkin Numbers ◽  
Recursive Formulas

scholarly journals Several Explicit and Recursive Formulas for the Generalized Motzkin Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Recursive Formula ◽  
Catalan Numbers ◽  
Explicit Formulas ◽  
Motzkin Numbers ◽  
Recursive Formulas

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

The digital function filters – algorithms and applications

2013 ◽  
Vol 61 (2) ◽  
pp. 371-377
Author(s):  
M. Siwczyński ◽  
A. Drwal ◽  
S. Żaba
Keyword(s):  
Digital Filters ◽  
Phase Shifters ◽  
Square Root ◽  
Real Power ◽  
The Real ◽  
Distributed Parameters ◽  
Digital Modeling ◽  
Analysis And Synthesis ◽  
Basic Functions ◽  
Recursive Formulas

Abstract The simple digital filters are not sufficient for digital modeling of systems with distributed parameters. It is necessary to apply more complex digital filters. In this work, a set of filters, called the digital function filters, is proposed. It consists of digital filters, which are obtained from causal and stable filters through some function transformation. In this paper, for several basic functions: exponential, logarithm, square root and the real power of input filter, the recursive algorithms of the digital function filters have been determined The digital function filters of exponential type can be obtained from direct recursive formulas. Whereas, the other function filters, such as the logarithm, the square root and the real power, require using the implicit recursive formulas. Some applications of the digital function filters for the analysis and synthesis of systems with lumped and distributed parameters (a long line, phase shifters, infinite ladder circuits) are given as well.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

Recursive formulas of a multiple-sectioned transmission line

1974 ◽  
Vol 21 (5) ◽  
pp. 640-642
Author(s):  
A. Hu ◽  
Foo Lam ◽  
Chiang Lin
Keyword(s):  
Transmission Line ◽  
Recursive Formulas

Recursive formulas for transforming ODEs to a polynomial form

1990 ◽  
Vol 15 (3) ◽  
pp. 233-238
Author(s):  
Benjamin Charny
Keyword(s):  
Polynomial Form ◽  
Recursive Formulas

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

2004 ◽  
Vol 14 (09) ◽  
pp. 3337-3345 ◽  
Author(s):  
JIANPING PENG ◽  
DUO WANG
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Sufficient Condition ◽  
First Order ◽  
Recursive Formulas ◽  
Unique Normal Forms

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

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