Triangle Geometry and Jacobstahl Numbers

Abstract

The convergence properties of certain triangle centres on the Euler line of an arbitrary triangle are studied. Properties of the Jacobsthal numbers, which appear in this process, are examined, and a new formula is given. A Ja- cobsthal decomposition of Pascal's triangle is presented. This review article takes as its motivation a simple problem in ele- mentary triangle geometry to study some properties of the Jacobs- thal numbers, defined by the recurrence relation an+2 = an+1 + 2an;a0 = 0;a1 = 1 (1) These numbers form the sequence 0;1;1;3;5;11;21;43;::: (Sloane, A001045). We let J(n) or Jn stand for the nth Jacobsthal number, starting with J(0)=0. These numbers are linked to the binomial co- ecients in a number of ways. Traditional formulas for