Dimensions and the probability of finding odd numbers in Pascal's triangle and its relatives

Frame, Michael; Neger, Nial

doi:10.1016/j.cag.2009.10.002

Cited by 4 publications

(1 citation statement)

References 9 publications

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“…In the case of r = 3 and 5, the Pascal-Sierpinski gaskets are similar to the corresponding delay plots. However, the Pascal-Sierpinski gaskets of modulo r = 4 and 6 exhibit disordered structure, and they looks different from the delay plots of such r. According to the theory [27] of the Pascal-Sierpinski gasket, the Pascal-Sierpinski gasket possess the same structure as the corresponding delay plot if r is a prime number; however, the pattern is more complicated if r is a composite number.…”

Section: Comparison With the Pascal-sierpinski Gasketsmentioning

confidence: 97%

Fractal behind smart shopping

Yamamoto,

Yamazaki

2011

Preprint

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The 'minimal' payment-a payment method which minimizes the number of coins in a purse-is presented. We focus on a time series of change given back to a shopper repeating the minimal payment. The delay plot shows visually that the set of successive change possesses a fine structure similar to the Sierpinski gasket. We also estimate effectivity of the minimal-payment method by means of the average number of coins in a purse, and conclude that the minimal-payment strategy is the best to reduce the number of coins in a purse. Moreover, we compare our results to the rule-60 cellular automaton and the Pascal-Sierpinski gaskets, which are known as generators of the discrete Sierpinski gasket.

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Section: Comparison With the Pascal-sierpinski Gasketsmentioning

confidence: 97%

Fractal behind smart shopping

Yamamoto,

Yamazaki

2011

Preprint

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Fractal behind coin-reducing payment

Yamamoto

Yamazaki

2012

Chaos, Solitons & Fractals

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Pascal's Prism

Brothers¹

2012

Math Gaz

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Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

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Dimensions and the probability of finding odd numbers in Pascal's triangle and its relatives

Cited by 4 publications

References 9 publications

Fractal behind smart shopping

Fractal behind smart shopping

Fractal behind coin-reducing payment

Pascal's Prism

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