Built on earlier works of Larcombe on a certain class of non-terminating expansions of the sine function, we set up two new \( {_{}{3}F_2} \) summation formulas via integration.

In this paper, we investigate exhaustively the cyclically indecomposable triple systems \( TS_\lambda(v) \) for \( \lambda = 2, v \leq 33 \) and \( \lambda = 3, v \leq 21 \), and we identify the decomposable ones. We also construct, by using Skolem-type and Rosa-type sequences, cyclically indecomposable two-fold triple systems \( TS_2(v) \) for all admissible orders. Further, we investigate exhaustively all cyclic \( TS_2(v) \) that are constructed by Skolem-type and Rosa-type sequences up to \( v \leq 45 \) for indecomposability.

We show that if the independence number of a graph is \( \alpha \), then the eternal security number of the graph is at most \( \binom{\alpha+1}{2} \), solving a problem stated by Goddard, Hedetniemi, and Hedetniemi \([JCMCC, \text{ vol. } 52, \text{ pp. } 160-180]\).

Let \( n \) be a natural number. We obtain convolution-type formulas for the total number of parts in all partitions of \( n \) of several different kinds.

In this paper, we establish a doubling method to construct inequivalent Hadamard matrices of order \( 2n \), from Hadamard matrices of order \( n \). Our doubling method uses heavily the symmetric group \( S_n \), where \( n \) is the order of a Hadamard matrix. We improve the efficiency of the method by introducing some group-theoretical heuristics. Using the doubling method in conjunction with the standard 4-row profile criterion, we have constructed several millions of new inequivalent Hadamard matrices of orders \(48, 56, 64, 72, 80, 88, 96,\) and several hundreds of inequivalent Hadamard matrices of orders 672 and 856. The Magma code segments, included in this paper, allow one to compute many more inequivalent Hadamard matrices of the above orders and all other orders of the form \( 8t \).

In this paper, we determine analytically the number of balanced, unlabelled, 3-member covers of an unlabelled finite set, which is then used to find the number of non-isomorphic optimal lottery sets of cardinality three. We also determine numerically the number of non-isomorphic optimal playing sets for lotteries in which a single correct number is required to win a prize.

A fire breaks out on a graph \( G \) and then \( f \) firefighters protect \( f \) vertices. At each subsequent interval, the fire spreads to all adjacent unprotected vertices, and firefighters protect \( f \) unburned vertices. Let \( f_G \) be the minimum number of firefighters needed to contain a fire on graph \( G \). If the triangular grid goes unprotected to time \( t = k \) when \( f_G \) firefighters arrive and begin protecting vertices, the fire can be contained by time \( t = 18k + 3 \) with at most \( 172k^2 + 58k + 5 \) vertices burned.

A construction is given for a Restricted Sarvate-Beam Triple System in the case \( v = 8 \). This is the extremal case, since a Restricted SB Triple System cannot exist for \( v > 8 \).

A \( t \)-\((v, k, \lambda) \) covering is a set of blocks of size \( k \) such that every \( t \)-subset of a set of \( v \) points is contained in at least \( \lambda \) blocks. The cardinality of the set of blocks is the size of the covering. The covering number \( C_\lambda(v, k, t) \) is the minimum size of a \( t \)-\((v, k, \lambda) \) covering. In this article, we find upper bounds on the size of \( t \)-\((v, k, 2) \) coverings for \( t = 3, 4 \), \( k = 5, 6 \), and \( v \leq 18 \). Twelve of these bounds are the exact covering numbers.