Some Properties of a Third Order Partial Recurrence Relation

This paper explores a connection between third order recursive sequences and generalized continued fractions by analogy with second order recursive sequences and ordinary two-dimensional continued fractions. It does this with a partial recurrence relation which is related to the original third order recurrence relation, and raises a related conjecture.

___

  • [1] Horadam, A.F., “Basic properties of a certain generalized sequence of numbers”, The Fibonacci Quarterly, 3: 161-176, (1965).
  • [2] Deveci, O. and Shannon, A.G., “Pell-Padovan-circulant sequences and their applications”, Notes on Number Theory and Discrete Mathematics, 23 (3): 100-114, (2017).
  • [3] Sloane, N.J.A., “The On-Line Encyclopedia of Integer Sequences”, http://oeis.org, (1964).
  • [4] Hildebrand, F.B., Introduction to Numerical Analysis, McGraw-Hill, New York, p.461, (1956).
  • [5] Feinberg, M., “New Slants”, The Fibonacci Quarterly, 2: 223-227, (1964).
  • [6] Jarden, D., Recurring Sequences, Jerusalem: Riveon Lematika, p.114, (1966).
  • [7] Shannon, A.G., Anderson, P.G. and Horadam, A.F., “Properties of Cordonnier, Perrin and Van der Laan numbers”, International Journal of Mathematical Education in Science and Technology, 37: 825-831, (2006).
  • [8] Shannon, A.G., Horadam, A.F. and Anderson, P.G., “The auxiliary equation associated with the Plastic number”, Notes on Number Theory and Discrete Mathematics, 12: 1-12, (2006).
  • [9] Shannon, A.G. and Wong, C.K., “Some Properties of Generalized Third Order Pell Numbers”, Congressus Numerantium, 201: 345-351, (2010).
  • [10] Gnanadoss, A.A., “Contracting Bernoulli’s iteration and recurrence relations”, The Mathematical Gazette, 44: 221-223, (1960).
  • [11] Anderson, P.G., Brown, T.C. and Shiue, P.J.-S., “A simple proof of a remarkable continued fraction identity”, Proceedings of the American Mathematical Society, 123:2005-2009, (1995).
  • [12] Anderson, P.G., “Notes and extensions for a remarkable continued fraction”, The Fibonacci Quarterly, 55: 9-14, (2017).
  • [13] Van der Cruyssen, P., “Linear difference equations and generalized continued fractions”, Computing, 22: 269-278, (1979).
  • [14] Szekeres, G., “Multidimensional Continued Fractions”, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 13: 113–140, (1970).
  • [15] Van der Cruyssen. P., “A continued fraction algorithm”, Numerische Mathematik, 37: 149-156, ( 1981).
  • [16] Bernstein, L., The Jacobi-Perron Algorithm: Its Theory and Application (Lecture Notes in Mathematics 207), Berlin: Springer, Ch.2, (1971).