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This paper addresses divisibility properties of some families of sequences arising from partial sums of a strong divisibility sequence. In particular, we demonstrate periodicity of greatest common divisor within 1-fibonacci numbers. We also present congruences within this sequence modulo a prime number p where p ≡ ±1 (mod 5) and p ≡ ±2 (mod 5).

The generalization of the integral formula with three nodes is introduced, and some sharp and the best possible inequalities for the functions whose higher order derivatives belong to L p spaces are given. We establish non-weighted version of the three point integral formula. From the general non-weighted formula we shall get the famous Simpson, dual Simpson and Maclaurin formulae. Some new errors of approximation in these integral formulae are obtained. (2010): 26D15, 65D30, 65D32. Mathematics subject classification