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. Present Address : Osamu Matsuda Tsuyama National College of Technology 624-1, Numa, Tsuyama-City, Okayama, Japan, 708-8509 e-mail : matsuda@tsuyama-ct.ac.jp