# The study of the generalized Pascal triangle by shwarma

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```									On generalization of Pascal's triangle
Osamu Matsuda (Tsuyama National College of Technology)

Suggested by Pascal’s triangle, a student of mine found some rules of the coefficients of
(x^2+x+1)^n or (x^3+x^2+x+1)^n. The coefficients of (x^k+…+1)^n may be called the
triangle of type P[k].
Numbers in the triangle of type P[2], i.e. Pascal's triangle, are sums of two consecutive
numbers on the up stair. Similarly, numbers in the triangle of type P[k] are sums of k
consecutive numbers on the up stair.
Next, generalizing the Fibonacci sequence f(n), we introduce k Fibonacci sequence f(n) by
f(n)=f(n-1)+…+f(n-k), f(1)=1, f(-m)=0, where m is any non negative integer. Then k
Fibonacci sequence is obtained from the sum of diagonal numbers in the triangle of type P[k].
Moreover, the limit value x of f(n)/f(n+1) as n tends to infinity satisfies the equation x^k+…
+x-1=0. In Particular, when k=2, x becomes the Golden ratio.
On the other hand, we found some patterns on the triangle of type P[k], for example, there
are triangular numbers or numbers of vertexes of (hyper) triangular pyramid on the triangle of
type P[k], etc.
Note that this study was done by four students in our college who are fifteen years old.
Recently, a student from Mongolia found interesting functions related to series with k
Fibonacci sequence coefficients. They are applied to the probability.
Generalized Pascal's triangles are useful in teaching. Various mathematical phenomena on
these triangles have been found.