On the k–Lucas Numbers of Arithmetic Indexes

DOI: 10.4236/am.2012.310175   PDF   HTML   XML   5,945 Downloads   8,262 Views   Citations

Abstract

In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

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S. Falcon, "On the k–Lucas Numbers of Arithmetic Indexes," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1202-1206. doi: 10.4236/am.2012.310175.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] S. Falcon, “On the k-Lucas Numbers,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 21, 2011, pp. 1039-1050
[2] S. Falcon and A. Plaza, “On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022
[3] S. Falcon and A. Plaza, “The k-Fibonacci Sequence and the Pascal 2-Triangle,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 38-49. doi:10.1016/j.chaos.2006.10.022
[4] S. Falcon and A. Plaza, “On k-Fibonacci Numbers of Arithmetic Indexes,” Applied Mathematics and Computation, Vol. 208, 2009, pp. 180-185 doi:10.1016/j.amc.2008.11.031

  
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