On Some Properties of Fibonacci Diagonals in Pascal's Triangle
نویسندگان
چکیده
Although it has been studied extensively, Pascal's triangle remains fascinating to explore and there always seems to be some new aspects that are revealed by looking at it closely. In this paper we shall examine a few nice properties of the so-called Fibonacci diagonals, that is, those slant lines whose entries sum to consecutive terms of the Fibonacci sequence. We adopt throughout our text the convention that the n^ Fibonacci diagonal is the one that contains the binomial coefficients
منابع مشابه
A Lucas Triangle Primality Criterion Dual to That of Mann-shanks
subject to the initial conditions A(l, 0) = 1, A(l, 1) ='2, with 4(n, /c) = 0 for & < 0 or k > n. This array has been called a Lucas triangle by Feinberg [1], because rising diagonals sum to give the Lucas numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ..., in contrast to the rising diagonals in the standard Pascal triangle where rising diagonals sum to give the Fibonacci numbers 1, 1, ...
متن کامل458 [Dec. PASCAL'S TRIANGLE MODULO
In "Mathematical Games" in the December 1966 issue of Scientific American, Martin Gardner made the following statement regarding Pascals triangle: "Almost anyone can study the triangle and discover more properties, but it is unlikely that they will be new, for what is said here only scratches the surface of a vast literature." But, of course, many new results have been discovered since 1966 and...
متن کاملSierpinski’s Triangle and the Prouhet-Thue-Morse Word
Sierpinski's triangle is a fractal and the Prouhet-Thue-Morse word is suÆciently chaotic to avoid cubes. Here we observe that there is at least a tenuous connection between them: the Sierpinski triangle is evident in Pascal's triangle mod 2 whose inverse, as an in nite lower-triangular matrix, involves the Prouhet-Thue-Morse word. Pascal's triangle mod 2 (Fig. 1b) is a discrete version of the f...
متن کاملGeometry of Binomial Coefficients
This note describes the geometrical pattern of zeroes and ones obtained by reducing modulo two each element of Pascal's triangle formed from binomial coefficients. When an infinite number of rows of Pascal's triangle are included, the limiting pattern is found to be "self-similar," and is characterized by a "fractal dimension" log2 3. Analysis of the pattern provides a simple derivation of the ...
متن کاملOn Some Number Sequences Related to the Parity of Binomial Coefficients
It is well known that striking patterns of triangles can be produced from Pascal's triangle by replacing each binomial coefficient by its residue with respect to a certain modulus. The arrays thus produced were considered by various authors; see, for instance, Gould [5], Gardner [1], Long [10], or Sved [17]. For example, Pascal's triangle mod 2 (Fig. 1) is the array of zeros and ones obtained b...
متن کامل