Combinatorially, each term R(N)x counts the R(N) partitions of N into distinct Fibonacci numbers. Some of the recursion properties of this sequence are investigated in [2]. The difficulties in producing this sequence are more computational than analytic in that usual generation methods quickly consume computer resources. Our major interest is in the related sequence 1, 3, 8, 16, 24, 37, ..., An^ whose rfi term is the least N such that n = R(N), emphasized in boldface in (1) above. The general term of this sequence (see [8]), designated A013583 in Sloane's' on-line database of sequences, is still unknown. The 330 terms found in this note almost triple the 112 terms reported by Shallit [8]. Carlitz [3, 4], Klamer [7], and Hoggatt [4, 6], among others, have studied the representation of integers as sums of Fibonacci numbers and particularly Zeckendorf representations. The Zeckendorf representation of a natural number N uses only positive-subscripted, distinct, and nonconsecutive Fibonacci numbers and is unique. We have used the Zeckendorf representation of N to write R(N) in [1] and [2].

In this note, we study exclusively n-length binary sequences where at least one 1 is adjacent to (or touching) another 1. For brevity, we refer to them as “binary sequences with some touching 1’s.” They are not to be confused with n-length binary sequences where every 1 is adjacent to at least one other 1 [1]. We let C n denote the collection with unrestricted content, and we let D n denote the subset of C n with even content. (Content is defined as the number of 1’s in a binary sequence.)

In the sub-title of his text, “Mathematics of Choice,” Niven [7] characterized the study of combinatorics as “How to count without counting,” to highlight the role of enumeration in combinatorics. While the importance of knowing “how many” cannot be denied, there are many instances where the unique names of members of an enumeration can serve as codes to perform external control tasks or even adaptively influence the future course of the enumeration. This is particularly true when the names are represented as ordered collections of integers. To form a next name from a present name, application of a suitable algorithm is the adaptive influence needed. To cite one example which we will expand in detail later, Even [1, page 32] introduces a position shifting algorithm for successively generating unique names of combinations C(n, m). In recent work on counting and naming conversations on crossbar systems [2–4], we developed an algorithm for finding the successive names of a structure we call cascaded combinations. (Our concept of cascaded combinations is fully described in a subsequent section.) Additionally, we needed a simple algorithm for naming multikind permutations of n. These are the familiar permutations of n with r 1 of the first kind, r 2 of the second kind, etc., up through r k of the kth (and last) kind. To our surprise, an extensive library search and inquiries of colleagues did not yield a statement of what should be a commonplace algorithm. We temporarily abandoned the search when we discovered we could use our algorithm for generating cascaded combinations as a very simple “guide” algorithm for generating the permutation names.

A very simple divide-and-conquer method for a class of word statements which breaks down a statement to obtain several simple state tables is presented. By following rules for combining the tables, a correct final state table can be constructed. The method has additional bonuses in that what if changes can be explored, and logically impossible word statements can be detected. The method is easy to apply and it provides students with a way of solving difficult problems correctly.< >

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.

Taylor expansions and subsequent arrangement in linear simultaneous-equation form lead to matrices for the numerical application of the bilinear transformation z=(s+1)/(s-1). The matrices are somewhat similar to those previously reported, but the proof seems more adaptable to the classroom.

Through use of vertex matrices and a path inversion-induced duality between paths and loops, it is shown that the work of finding additional graph formulas from a given formula can often be reduced.

A method for generating restricted partitions of numbers is presented as an aid in helping the student construct three-arc cycles in certain directed graphs. Through this and similar devices the study of linear graph theory, as a whole, becomes more real to the student.

The existing symbolic method for the partial inversion of coefficient matrices in a system of linear equations does not cover all possible cases. The method is extended here to include the missing cases, and some classroom simplifications of the original manipulation rules are introduced.

To a student in linear graph theory, many concepts which later become almost second nature often need at first step-by-step, concrete demonstrations. For example, cut sets and cut vertices become increasingly meaningful if the student is given a procedure whereby he, by himself, can correctly and confidently identify all cut sets and cut vertices in any given graph no matter how complex. For this purpose, an earlier definition of ambits is extended to provide the basis for a combinatorial method for finding all cut sets and cut vertices of a connected graph. As by-products, interesting properties of ambits are disclosed.

There are surprisingly few graphs which are both regular (same degree of incidence at each node) and have regular duals. Because these graphs all can be arranged to appear as regular geometric solids, they are known, specifically, as Platonic bodies. While direct application of graph theoretical methods can be used to discover all the graphs, this paper shows how topological and electrical restrictions also can be used to find the Platonic bodies through graphs of resultant resistance networks.