**Volume 19** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Professor Anthony Shannon at F_{4} × F_{5} × F_{5} years**

*Editorial. Pages 1–4*

Krassimir T. Atanassov

Editorial (PDF, 84 Kb)

**The Pascal–Fibonacci numbers**

*Original research paper. Pages 5–11*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 144 Kb) | Abstract

**Pulsating Fibonacci sequence**

*Original research paper. Pages 12–14*

Krassimir T. Atanassov

Full paper (PDF, 100 Kb) | Abstract

**A note on modified Jacobsthal and Jacobsthal–Lucas numbers**

*Original research paper. Pages 15–20*

Julius Fergy T. Rabago

Full paper (PDF, 144 Kb) | Abstract

**Relations on Jacobsthal numbers**

*Original research paper. Pages 21–23*

S. Arunkumar, V. Kannan and R. Srikanth

Full paper (PDF, 120 Kb) | Abstract

**New explicit representations for the prime counting function**

*Original research paper. Pages 24–27*

Mladen Vassilev-Missana

Full paper (PDF, 148 Kb) | Abstract

(where σ is the sum-of-divisor function and ψ is the Dedekind’s function) are proposed and proved. Also a general theorem (Theorem 1) is obtained that gives infinitely many explicit formulae for the prime counting function π (depending on arbitrary arithmetic function with strictly positive values, satisfying certain condition).

**On the area and volume of a certain regular solid and the Diophantine equation 1/2 = 1/ x + 1/y + 1/z**

*Original research paper. Pages 28–32*

Julius Fergy T. Rabago and Richard P. Tagle

Full paper (PDF, 147 Kb) | Abstract

**BBP-type formulas, in general bases, for arctangents of real numbers**

*Original research paper. Pages 33–54*

Kunle Adegoke and Olawanle Layeni

Full paper (PDF, 221 Kb) | Abstract

**On subsets of finite Abelian groups without non-trivial solutions of x_{1 }+ x_{2} + … + x_{s} – sx_{s}_{+1} = 0**

*Original research paper. Pages 55–59*

Ran Ji and Craig V. Spencer

Full paper (PDF, 202 Kb) | Abstract

_{1}+ … + x

_{s}− sx

_{s+1}= 0 with x

_{i}∈ A(1 ≤ i ≤ s + 1). Let

where rk(H) is the rank of H. We prove that for any n ∈ ℕ, , where is a fixed constant depending only on s.

**A note on the modified q-Dedekind sums**

*Original research paper. Pages 60–65*

Serkan Araci, Erdoğan Şen and Mehmet Acikgoz

Full paper (PDF, 184 Kb) | Abstract

*p*-adic continuous function for an odd prime to inside a

*p*-adic

*q*-analogue of the higher order Extended Dedekind-type sums related to

*q*-Genocchi polynomials with weight

*α*by using fermionic

*p*-adic invariant

*q*-integral on ℤ

_{p}.

**Modular zero divisors of longest exponentiation cycle**

*Original research paper. Pages 66–69*

Amin Witno

Full paper (PDF, 115 Kb) | Abstract

*w*mod

^{k}*n*, given that gcd(

*w*,

*n*) > 1, can reach a maximal cycle length of

*ϕ*(

*n*) if and only if

*n*is twice an odd prime power,

*w*is even, and

*w*is a primitive root modulo

*n*=2.

**A characterization of canonically consistent total signed graphs**

*Original research paper. Pages 70–77*

Deepa Sinha and Pravin Garg

Full paper (PDF, 164 Kb) | Abstract

_{σ}(v) = Π

_{ej ∈ Ev}, where E

_{v}is the set of edges e

_{j}incident at v in S. If S is canonically marked, then a cycle Z in S is said to be canonically consistent (C-consistent) if it contains an even number of negative vertices and the given sigraph S is C-consistent if every cycle in it is C-consistent. The total sigraph T(S) of a sigraph S = (V, E, σ) has T(S

^{e}) as its underlying graph and for any edge uv of T(S

^{e}),

In this paper, we establish a characterization of canonically consistent total sigraphs.

**Integer sequences from walks in graphs**

*Original research paper. Pages 78–84*

Ernesto Estrada and José A. de la Peña

Full paper (PDF, 168 Kb) | Abstract

_{j}(N) = N

^{0}– N

^{1}+ N

^{2}– … + N

^{2j}and E

^{j}(N) = –N

^{0}+ N

^{1}– N

^{2}+ … + N

^{2j+1}(j = 0, 1, 2, …) and the corresponding integer sequences. We prove that these integer sequences, e.g., S

_{0}(N) = O

_{0}(N), O

_{1}(N), …, O

_{r}(N), … and S

_{E}(N) = E

_{0}(N), E

_{1}(N), …, E

_{r}(N), … correspond to the number of odd and even walks in complete graphs KN. We then prove that there is a unique family of graphs which have exactly the same sequence of odd walks between connected nodes and of even walks between pairs of nodes at distance two, respectively. These graphs are the crown graphs: G

_{2n}= K

_{2}⊗ K

_{n}.