Topic
Pascal's pyramid
About: Pascal's pyramid is a research topic. Over the lifetime, 6 publications have been published within this topic receiving 5 citations.
Papers
1 Jan 1996
TL;DR: In this article, a star S is a center covering star w.r.t. LCM if it covers its center in the above sense, and it is minimal if it does not contain any such center covering stars with the same center which is a proper subset of S.
Abstract: Take an entry X inside m-dimensional Pascal’s pyramid consisting of m-nomial coefficients. We call a set of successive r entries starting A on the half line XA a ray of length r with center X, where A is one of the m(m + 1) entries surrounding X, and the union of nonempty set of rays with center X a star with center X. When, in the following, we assume that a star S is translatable in parallel with its center in Pascal’s pyramid, we sometimes use the same word “star” instead of “star configuration” for brevity. For a configuration of two sets of entries U and V in the pyramid, we say that U covers V w.r.t. LCM if the equality LCM U ⋃ V = LCM U always holds independent of the location of U and V as long as they are contained in the pyramid and their relative location is unchanged. We say that a star S is a center covering star w.r.t. LCM if it covers its center in the above sense, and it is minimal if it does not contain any such center covering star with the same center which is a proper subset of S.
3 citations
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2 citations
1 Jan 2019
TL;DR: In this paper, the authors present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle.
Abstract: Many properties have been found hidden in Pascal’s triangle. In this paper, we will present several known properties in Pascal’s triangle as well as the properties that lift to different extensions of the triangle, namely Pascal’s pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.
1 citations
Posted Content•
TL;DR: The tri-atomic sequence as discussed by the authors is a two-dimensional Pascal with memory sequence, which is a generalization of Stern's diatomic sequence 0,1, 1, 2, 1/3, 2/3/4, 3/4/5, 4/5/6, 6/7, 8/8, 9/10, 11/12, 12/13, 14/15, 16/16, 17/18, 18/19, 20/20, 21/21
Abstract: Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation A_2n=A_n and A_{2n+1} = a_n + a_{n+1}, where A_0=0 and A_1=1), which has long been known. Using a particular multi-dimensional continued fraction algorithm (the Farey algorithm), we will generalize the diatomic sequence to a collection of numbers that quite naturally should be called the tri-atomic sequence (or a two-dimensional Pascal with memory sequence). As continued fractions and the diatomic sequence can be thought of as coming from systematic subdivisions of the unit interval, this new tri-atomic sequence will arise by a systematic subdivision of a triangle. We will discuss some of the algebraic properties for the tri-atomic sequence.
TL;DR: Notation for levels and coordinates of elements, a standard algorithm for generating the values of various elements, and a ratio method that is not dependent on the calculation of previous levels are discussed.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2012 | 1 |
| 1998 | 1 |
| 1996 | 1 |
| 1991 | 1 |
| 1968 | 1 |