Fibonacci, and Lucas Pascal Triangles

Fibonacci words in hyperbolic Pascal triangles

The hyperbolic Pascal triangle HPT 4,q (q ≥ 5) is a new mathematical construction, which is a geometrical generalization of Pascal’s arithmetical triangle. In the present study we show that a natural pattern of rows of HPT 4,5 is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the h...

Matrix Powers of Column-justified Pascal Triangles and Fibonacci Sequences

It is known that if Ln, respectively Rn9 are n x n matrices with the (/, j ) * entry the binomial coefficient (y~l)? respectively (^l)), then Ln = In (mod 2), respectively R„=In (mod 2), where In is the identity matrix of dimension n>\ (see, e.g., Problem PI 073 5 in the May 1999 issue of Arner. Math Monthly). The entries of Ln form a left-justified Pascal triangle and the entries of Rn result ...

Tea Tasting and Pascal Triangles

ON LUCAS v-TRIANGLES

are well known. A list of such basic identities can be found in [3]. If A ^ ±1 or B ^ 1, then w1? s^,... are nonzero by [1], and so are vx = u2lul9 v2 = M4/M2, ... . In the case A = B 1, we noted in [1] that un = 0 o 31n. IF vw = 0, then uln = i/wvw = 0; hence, 31n and un = Q, which is impossible since v~Au = 4B (cf. [3]). Thus, v0,v1? v2,... are all nonzero. We set vw! = Ilo ^ reg...

FIBONACCI ^-SEQUENCES, PASCAL-r TRIANGLES, AND A—IN-A-ROW PROBLEMS

In what follows, we use the Fibonacci sequences of order k9 as for example in Philippou and Muwafi [2] (although modified somewhat here), and the PascalT triangles, as in Turner [6], to solve a number of enumeration problems involving the number of binary numbers of length n which have (or do not have) a string of k consecutive ones, subject to various auxilliary conditions (no k consecutive on...