Band 11, Heft 1

We give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.

Let g j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x 1 , . . . , x n }. We show that for any k > 0, and n = 5, the quantity g 0 – g k is unbounded. Furthermore, we provide examples with g 0 > g k for n ≥ 6 and g 0 > g 1 for n ≥ 4.

We reintroduce an interpretation of the k th-fold self convolution of the Catalan numbers by showing that they count the number of words in symbols X and Y , where the total number of Y 's is k more than the total number of X 's, and at no time are there more Y 's than k plus the number of X 's. Using this, we exhibit some of the wide variety of combinatorial interpretations of the k th-fold self convolution of the Catalan numbers. Finally, we show how these numbers appear as the last column in a truncated Pascal's triangle.

Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.

For any set T = { t 1 , t 2 , . . . , t n } ⊆ ℕ, a basis A is said to have a T -order s if every sufficiently large integer is the sum of s – t 1 or s – t 2 or . . . or s – t n elements taken from A (allowing repetitions), where s is the least integer with this property. We write ord ( T ) ( A ) = s . In this paper, we characterize those bases A which have a T -order.

Independently, Pirillo and Varricchio, Halbeisen and Hungerbühler and Freedman considered the following problem, open since 1992: Does there exist an infinite word w over a finite subset of ℤ such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden's theorem on arithmetic progressions.

A ternary inclusion-exclusion polynomial is a polynomial of the form where p, q , and r are integers ≥ 3 and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting p, q , and r to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of Q { p,q,r } and Q { p,q,s } , with r ≡ s (mod pq ). More specifically, we consider the case where 1 ≤ s < max( p, q ) < r , and obtain a recursive estimate for the function A ( p, q, r ) – the function that gives the maximum of the absolute values of the coefficients of Q { p,q,r } . A simple corollary of our main result is the following absolute estimate. If s ≥ 1 and r ≡ ± s (mod pq ), then A ( p, q, r ) ≤ s .