Blocks within the period of Lucas sequence

Rima P. Patel, Dr. Devbhadra V. Shah

Abstract


In this paper, we consider the periodic nature of the sequence of  Lucas numbers L_n defined by the recurrence relation L_n= L_(n-1)+L_(n-2); for all n≥2; with initial condition L_0=2 and L_1=1. For any modulo m>1, we introduce the ‘blocks’ within this sequence by observing the distribution of residues within a single period of Lucas sequence. We show that length of any one period of the Lucas sequence contains either 1,2 or 4 blocks. 


Keywords


Fibonacci sequence, Lucas sequence, Periodicity of Lucas sequence

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References


Kramer J., Hoggatt. Jr. V. E., Special cases of Fibonacci Periodicity, Fibonacci Quarterly, 1:5, 1972, 519 – 522.

Marc R., Properties of the Fibonacci sequence under various moduli, Master’s Thesis, Wake Forest University, 1996.

Patel R. P., Shah D. V., Periodicity of generalized Lucas numbers and the length of its period under modulo 2^e, Mathematics Today, 33, 2017, 67 – 74.

Robinson D. W., The Fibonacci matrix modulo m, The Fibonacci Quarterly, 1, 29 – 36, 1963.

Thomas K., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Inc., New York, 2001.

Wall D. D., Fibonacci series modulo m, The American Mathematical Monthly, 67, 1960, 525 – 532.




DOI: http://dx.doi.org/10.23755/rm.v41i0.664

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Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.