An Investigation of Sequences Derived from Hoggatt Sums and Hoggatt Triangles

Daniel C. Fielder and Cecil O. Alford

A chapter in Applications of Fibonacci Numbers, 1990, pp 77-88 from Springer

Abstract: Abstract In a recent note [1], the authors discuss derivations of integer sequences called Hoggatt Sums and associated triangular arrays called Hoggatt Triangles. The nomenclature was proposed as a tribute to the late Verner Hoggatt, Jr. since the investigation and extension of an unpublicized conjecture of Hoggatt ultimately resulted in the above sums and triangles. In personal correspondence [2], Hoggatt conjectured that the third (counting as 0, 1, 2, 3, …) right diagonal of Pascal’s triangle could be used to determine the sequence of integers, S0, S1, S2, … S m ,…, which are identically the Baxter permutation counts [3] of indices 0, 1, 2, …, m, … Hoggatt based his calculation algorithm for S m on sums of products between third diagonal terms from Pascal’s triangle and appropriately corresponding terms from a completed S m -1. The authors’ note [1] supplied the missing proof of Hoggatt’s conjecture. Hoggatt’s conjecture was then extended to include all right Pascal triangle diagonals indexed as 0, 1, 2, 3, …, d, …. For each d, the set of S m ’s became Hoggatt sums of order d, and the individual integers which sum to a particular S m became row members of a triangular array called a Hoggatt triangle of order d. With the inclusion of d as a variable parameter, the numerical results of [1] can be interpreted as sequences of (S d ) m ’s with Fixed index d and variable index m. For example, the Baxter permutation count values are Hoggatt sums of order three whose general sequence term is (S3) m . Sequences of Hoggatt sums follow a linear recursion which is index-variant in m, i.e., the calculation of (S d ) m for d fixed depends not only on previous members of the sequence but also depends on the value of m. Difference equations for this type of recursion are known to be difficult, if not impossible, to obtain by operational methods [4].

Keywords: Computer Algebra System; Position Sequence; Laurent Expansion; Rectangular Enclosure; Recursive Sequence (search for similar items in EconPapers)
Date: 1990
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-1910-5_8

Ordering information: This item can be ordered from
http://www.springer.com/9789400919105

DOI: 10.1007/978-94-009-1910-5_8

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().