On Families of Nonlinear Recurrences Related to Digits
Abstract
Consider the sequence of positive integers (u_n)_{n>=1} defined by u_1=1 and u_{n+1}=floor( sqrt(2)*(u_n+1/2)). Graham and Pollak discovered the unexpcted fact that u_{n+1}-2u_{2n-1} is just the n-th digit in the binary expansion of sqrt(2)E Fix w element R_{>0}. In this note, we first give two infinite families of similar nonlinear recurrences such taht u_{2n+1}-2u_{2n-1} indicates the n-th binary digit of w. Moreover, for all g>=2, we establish a recurrence such that u_{2n+1}-gu{2n-1} denotes the n-th digit w in the g-ary digital expansion.
- Publication:
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Journal of Integer Sequences
- Pub Date:
- May 2005
- Bibcode:
- 2005JIntS...8...32S
- Keywords:
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- Number Theory