In this note, we study Fibonacci-like sequences that are defined by the recurrence Sk = a, Sk+1 = b, Sn+2 ≡ Sn+1 + Sn (mod n + 2) for all n ≥ k, where k, a, b ∈ N, 0 ≤ a < k, 0 ≤ b < k + 1, and (a, b) 6= (0, 0). We will show that the number α = 0.SkSk+1Sk+2 · · · is irrational. We also propose a conjecture on the pattern of the sequence {Sn}n≥k.

FEASIBILITY OF INTEGER KNAPSACKS∗ ISKANDER ALIEV† AND MARTIN HENK‡ Abstract. Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R≥0 : Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number ass...

Let be the number of ordered pairs of paths in the plane, with unit steps E or N, that intersect k times in which the first path ends at the point (r,n-r) and the second path ends at the point (s,n-s). Let and We study the numbers , , , and , prove several simple relations among them, and derive a simpler formula for than appears in [1].

For given finite (unordered) graphs G and H, we examine the existence of a Ramsey graph F for which the strong Ramsey arrow F −→ (G)r holds. We concentrate on the situation when H is not a complete graph. The set of graphs G for which there exists an F satisfying F −→ (G)2 2 (P2 is a path on 3 vertices) is found to be the union of the set of chordal comparability graphs together with the set of...

SB (a, b) = { √ b2−a2 cos−1(a/b) , a < b , √ a2−b2 cosh−1(a/b) , a > b . In this paper, we find the greatest values α1, α2, α3 and α4, and the least values β1, β2, β3 and β4 such that the double inequalities α1SAH(a, b) + (1 − α1)C(a, b) < A(a, b) < β1SAH(a, b) + (1 − β1)C(a, b), α2SHA(a, b) + (1 − α2)C(a, b) < A(a, b) < β2SHA(a, b) + (1 − β2)C(a, b), α3SCA(a, b) + (1 − α3)H(a, b) < A(a, b) < β...

Let [n] denote an n-set. A subset,) of [n] i from j if i and j t/:. 5'. A collection of k-sets n called (n, k) completely if, for each ordered pair j) E [n] x [71,] with i n which i from j. Let denote the size of a smallest (71" k) completely Amongst other things, it will be shown that R( 71" k) for n > except when n (Hl) 1, and R(71" k) k + 1 for G) n k 2 /2. These results build on and extend ...

We consider the partially ordered set ([k]", <), which is defined as n-th product of the chain [k] = {0,1, 2 . . . . . k 1}, and study pairs (A, B) of incomparable sets A, B ~ [k]", that is, a :g b, a :~ b for all a ~ A, b ~ B or (in short notation) A : :~ B. We are concerned with the growth of the functions f,: {0, 1 . . . . . k ~} ---, {0, 1 . . . . . k"}, n ~ N, defined by f,(ct) = max{IN: A...

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the 3rst one, we say that the 3rst square is r-self-orthogonal, denoted by r-SOLS(n). It has been proved that the necessary condition for the existence of an r-SOLS(n) is n6 r6 n and r ∈ {n + 1; n − 1}. Zhu and Zhang conjectured that there is a...

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optim...

This paper generalizes the results in Aswal et al. (2003) on dictatorial domains. This is done in two ways. In the first, the notion of connections between pairs of alternatives in Aswal et al. (2003) is weakened to weak connectedness. This notion requires the specification of four preference orderings for every alternative pair. Domains that are linked in the sense of Aswal et al. (2003) with ...