On Sums of Products of Horadam Numbers

• Cerin, Zvonko
• Accepted : 2009.05.26
• Published : 2009.09.30
• 38 5

Abstract

In this paper we give formulae for sums of products of two Horadam type generalized Fibonacci numbers with the same recurrence equation and with possibly different initial conditions. Analogous improved alternating sums are also studied as well as various derived sums when terms are multiplied either by binomial coefficients or by members of the sequence of natural numbers. These formulae are related to the recent work of Belbachir and Bencherif, $\v{C}$erin and $\v{C}$erin and Gianella.

Keywords

Horadam generalized Fibonacci numbers;Lucas numbers;sums of products

References

1. H. Belbachir and F. Bencherif, Sums of products of generalized Fibonacci and Lucas numbers, Ars Combinatoria, (to appear).
2. Z. Cerin, Sums of products of generalized Fibonacci and Lucas numbers, Demonstratio Mathematica, 42(2) (2009), 247-258.
3. Z. Cerin, Properties of odd and even terms of the Fibonacci sequence, Demonstratio Mathematica, 39(1) (2006), 55-60.
4. Z. Cerin, On sums of squares of odd and even terms of the Lucas sequence, Proccedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 194(2009), 103-107.
5. Z. Cerin, Some alternating sums of Lucas numbers, Central European Journal of Mathematics 3(1) (2005), 1-13. https://doi.org/10.2478/BF02475651
6. E. Lucas, Theorie des Fonctions Numeriques Simplement Periodiques, American Journal of Mathematics 1(1878), 184-240. https://doi.org/10.2307/2369308
7. Z. Cerin, Alternating Sums of Fibonacci Products, Atti del Seminario Matematico e Fisico dell'Universita di Modena e Reggio Emilia, 53(2005), 331-344.
8. Z. Cerin and G. M. Gianella, On sums of squares of Pell-Lucas numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6(2006), A15.
9. Z. Cerin and G. M. Gianella, Formulas for sums of squares and products of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis., 140(2006), 113-122.
10. Z. Cerin and G. M. Gianella, On sums of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis., 141(2007), 23-31.
11. A. F. Horadam, Generating Functions for Powers of a Certain Generalized Sequence of Numbers, Duke. Math. J., 32(1965), 437-446. https://doi.org/10.1215/S0012-7094-65-03244-8
12. N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/-njas/sequences/.