On a Conjecture by Hoggatt with Extensions to Hoggatt Sums and Hoggatt Triangles
نویسندگان
چکیده
In letters [1] to one of us (Fielder) in mid-1977, the late Verner Hoggatt conjectured that the third diagonal of Pascal's triangle could be used in a simple algorithm to generate rows of integers whose row sums equaled correspondingly indexed Baxter permutation values (see [3], [4]). Later, in 1978, Chung, Graham, Hoggatt, and Kleiman produced a remarkable paper [2] in which they derived a general solution for Baxter permutation values. In planning an extension of Hoggatts work, we searched for, but never found, a proof of Hoggatts conjecture or even a documented statement of the conjecture. Reference [2] did, however, state that Hoggatt had found a simple way of finding the first ten Baxter permutation values but, again, without giving the conjecture. In this note, we formalize Hoggatts conjecture, derive formulas for the values predicted by the conjecture, and then prove the conjecture. As new material, we extend Hoggatts conjecture to all Pascal diagonals. In so doing, we will introduce structures called Hoggatt triangles and integers called Hoggatt sums. These names were the explicit choice of one of us (Fielder) as a tribute to Verner Hoggatt for his work with Pascal triangles and, in some small way, to express gratitude for Vern's guidance, help, and friendship through the years. Finally, we report briefly on a computeraided experiment to obtain recursion formulas for selected Hoggatt sums.
منابع مشابه
Rises, Levels, Drops and “+” Signs in Compositions: Extensions of a Paper by Alladi and Hoggatt
COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Department of Mathematics, California State University Los Angeles 5151 State University Drive, Los Angeles, CA 90032-8204 [email protected] P. Z. Chinn Department of Mathematics, Humboldt State University, Arcata, CA 95521 [email protected] R. P. Grimaldi Department of Mathematics, Rose-Hulman Institute of Techno...
متن کاملFibonacci Identities as Binomial Sums II
As in [2], our goal in this article is to write some more prominent and fundamental identities regarding Fibonacci numbers as binomial sums.
متن کاملOn a Hoggatt-bergum Paper with Totient Function Approach for Divisibility and Congruence Relations
During their discussion of divisibility and congruence relations of the Fibonacci and Lucas numbers, Hoggatt & Bergum found values of n satisfying the congruences Fn E 0 (mod ft) or Ln E 0 (mod ft) . In this connection, Hoggatt & Bergum's research appears in Theorems 1, 3, 5, 6, and 7 of [4]. The present paper originated on the same lines in search of values of n that satisfy <$>(Fn) E 0 (mod f...
متن کاملA proof of the Hoggatt-Bergum conjecture
It is proved that if k and d are positive integers such that the product of any two distinct elements of the set {F2k, F2k+2, F2k+4, d} increased by 1 is a perfect square, than d has to be 4F2k+1F2k+2F2k+3. This is a generalization of the theorem of Baker and Davenport for k = 1.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1989