[seqfan] Re: Interesting set of palindromes

Mon Oct 24 03:00:07 CEST 2011

Note that b^n + 1/b^n = L(2n) (where L(n) = A000032(n) are the Lucas 
numbers). According to Zeckendorf, every integer can be represented 
uniquely as a sum of non-adjacent Lucas numbers where L(0) and L(2) can't 
both be used. Since L(2n+1) = L(2n+2) - L(2n), this representation can be 
transformed to a representation (not necessarily the only one) by sums of 
b^n + 1/b^n with coefficients in {-1,0,1}. Thus

1 = L(1) = L(2) - L(0)               = b^1 - 2 b^0 + b^(-1)
2 = L(0)                             = 2 b^0
3 = L(2)                             = b^1 + b^(-1)
4 = L(3) = L(4) - L(2)               = b^2 - b^1 - b^(-1) + b^(-2)
5 = L(1) + L(3) = L(4) - L(0)        = b^2 - 2 b^0 + b^(-2)
6 = L(0) + L(3) = L(4) - L(2) + L(0) = b^2 - b^1 + 2 b^0 - b^(-1) + b^(-2)
7 = L(4)                             = b^2 + b^(-2)
8 = L(1) + L(4) = L(4) + L(2) - L(0) = b^2 + b^1 - 2 b^0 + b^(-1) + b^(-2)

etc

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada

On Oct 19 2011, Dale Gerdemann wrote:

>Hello Seqfans,
>
> Here's an interesting set of palindromes that I uncovered. I wonder if 
> there are other such sets.
>
>
> Let b=(3+sqrt(5))/2. And let a(n) be A001906(n), offset so that a(-1) = 
> 0, a(0) = 1, a(1) = 3, etc
>
>
>1 = 1 = a(0)
>
>2 = 2 = 2a(0)
>
>3 = b^1 + b^-1 = a(1) + a(-1)
>
>4 = b^1 + 1 + b^-1 = a(1) + a(0) + a(-1)
>
>5 = b^1 + 2 + b^-1 = etc (a-forms are palindromes just like b forms)
>
>6 = 2b^1 + 2b^-1
>
>7 = b^2 + b^-2
>
>8 = b^2 + 1 + b^-2
>
>9 = b^2 + 2 + b^-2
>
>10 = b^2 + b^1 + b^-1 + b^-2
>
>11 = b^2 + b^1 + 1 + b^-1 + b^-2
>
>12 = b^2 + b^1 + 2 + b^-1 + b^-2
>
>13 = b^2 + 2b^1 + 2b^-1 + b^-2
>
>14 = 2b^2 + 2b^-2
>
>15 = 2b^2 + 1 + 2b^-2
>
>16 = 2b^2 + 2 + 2b^-2
>
>17 = 2b^2 + b^1 + b^-1 + 2b^-2
>
>18 = b^3 + b^-3
>
>19 = b^3 + 1 + b^-3
>
>20 = b^3 + 2 + b^-3
>
>21 = b^3 + b^1 + b^-1 + b^-3
>
>22 = b^3 + b^1 + 1 + b^-1 + b^-3
>
>23 = b^3 + b^1 + 2 + b^-1 + b^-3
>
>24 = b^3 + 2b^1 + 2b^-1 + b^-3
>
>25 = b^3 + b^2 + b^-2 + b^-3
>
>26 = b^3 + b^2 + 1 + b^-2 + b^-3
>
>27 = b^3 + b^2 + 2 + b^-2 + b^-3
>
>28 = b^3 + b^2 + b^1 + b^-1 + b^-2 + b^-3
>
>29 = b^3 + b^2 + b^1 + 1 + b^-1 + b^-2 + b^-3
>
>30 = b^3 + b^2 + b^1 + 2 + b^-1 + b^-2 + b^-3
>
>31 = b^3 + b^2 + 2b^1 + 2b^-1 + b^-2 + b^-3
>
>32 = b^3 + 2b^2 + 2b^-2 + b^-3
>
>33 = b^3 + 2b^2 + 1 + 2b^-2 + b^-3
>
>34 = b^3 + 2b^2 + 2 + 2b^-2 + b^-3
>
>35 = b^3 + 2b^2 + b^1 + b^-1 + 2b^-2 + b^-3
>
>36 = 2b^3 + 2b^-3
>
>37 = 2b^3 + 1 + 2b^-3
>
>38 = 2b^3 + 2 + 2b^-3
>
>39 = 2b^3 + b^1 + b^-1 + 2b^-3
>
>40 = 2b^3 + b^1 + 1 + b^-1 + 2b^-3
>
>41 = 2b^3 + b^1 + 2 + b^-1 + 2b^-3
>
>42 = 2b^3 + 2b^1 + 2b^-1 + 2b^-3
>
>43 = 2b^3 + b^2 + b^-2 + 2b^-3
>
>44 = 2b^3 + b^2 + 1 + b^-2 + 2b^-3
>
>45 = 2b^3 + b^2 + 2 + b^-2 + 2b^-3
>
>46 = 2b^3 + b^2 + b^1 + b^-1 + b^-2 + 2b^-3
>
>47 = b^4 + b^-4
>
>48 = b^4 + 1 + b^-4
>
>49 = b^4 + 2 + b^-4
>
>50 = b^4 + b^1 + b^-1 + b^-4
>
>51 = b^4 + b^1 + 1 + b^-1 + b^-4
>
>52 = b^4 + b^1 + 2 + b^-1 + b^-4
>
>53 = b^4 + 2b^1 + 2b^-1 + b^-4
>
>54 = b^4 + b^2 + b^-2 + b^-4
>
>55 = b^4 + b^2 + 1 + b^-2 + b^-4
>
>56 = b^4 + b^2 + 2 + b^-2 + b^-4
>
>57 = b^4 + b^2 + b^1 + b^-1 + b^-2 + b^-4
>
>58 = b^4 + b^2 + b^1 + 1 + b^-1 + b^-2 + b^-4
>
>59 = b^4 + b^2 + b^1 + 2 + b^-1 + b^-2 + b^-4
>
>60 = b^4 + b^2 + 2b^1 + 2b^-1 + b^-2 + b^-4
>
>61 = b^4 + 2b^2 + 2b^-2 + b^-4
>
>62 = b^4 + 2b^2 + 1 + 2b^-2 + b^-4
>
>63 = b^4 + 2b^2 + 2 + 2b^-2 + b^-4
>
>64 = b^4 + 2b^2 + b^1 + b^-1 + 2b^-2 + b^-4
>
>65 = b^4 + b^3 + b^-3 + b^-4
>
>66 = b^4 + b^3 + 1 + b^-3 + b^-4
>
>67 = b^4 + b^3 + 2 + b^-3 + b^-4
>
>68 = b^4 + b^3 + b^1 + b^-1 + b^-3 + b^-4
>
>69 = b^4 + b^3 + b^1 + 1 + b^-1 + b^-3 + b^-4
>
>70 = b^4 + b^3 + b^1 + 2 + b^-1 + b^-3 + b^-4
>
>71 = b^4 + b^3 + 2b^1 + 2b^-1 + b^-3 + b^-4
>
>72 = b^4 + b^3 + b^2 + b^-2 + b^-3 + b^-4
>
>73 = b^4 + b^3 + b^2 + 1 + b^-2 + b^-3 + b^-4
>
>74 = b^4 + b^3 + b^2 + 2 + b^-2 + b^-3 + b^-4
>
>75 = b^4 + b^3 + b^2 + b^1 + b^-1 + b^-2 + b^-3 + b^-4
>
>76 = b^4 + b^3 + b^2 + b^1 + 1 + b^-1 + b^-2 + b^-3 + b^-4
>
>77 = b^4 + b^3 + b^2 + b^1 + 2 + b^-1 + b^-2 + b^-3 + b^-4
>
>78 = b^4 + b^3 + b^2 + 2b^1 + 2b^-1 + b^-2 + b^-3 + b^-4
>
>79 = b^4 + b^3 + 2b^2 + 2b^-2 + b^-3 + b^-4
>
>80 = b^4 + b^3 + 2b^2 + 1 + 2b^-2 + b^-3 + b^-4
>
>81 = b^4 + b^3 + 2b^2 + 2 + 2b^-2 + b^-3 + b^-4
>
>82 = b^4 + b^3 + 2b^2 + b^1 + b^-1 + 2b^-2 + b^-3 + b^-4
>
>83 = b^4 + 2b^3 + 2b^-3 + b^-4
>
>84 = b^4 + 2b^3 + 1 + 2b^-3 + b^-4
>
>85 = b^4 + 2b^3 + 2 + 2b^-3 + b^-4
>
>86 = b^4 + 2b^3 + b^1 + b^-1 + 2b^-3 + b^-4
>
>87 = b^4 + 2b^3 + b^1 + 1 + b^-1 + 2b^-3 + b^-4
>
>88 = b^4 + 2b^3 + b^1 + 2 + b^-1 + 2b^-3 + b^-4
>
>89 = b^4 + 2b^3 + 2b^1 + 2b^-1 + 2b^-3 + b^-4
>
>90 = b^4 + 2b^3 + b^2 + b^-2 + 2b^-3 + b^-4
>
>91 = b^4 + 2b^3 + b^2 + 1 + b^-2 + 2b^-3 + b^-4
>
>92 = b^4 + 2b^3 + b^2 + 2 + b^-2 + 2b^-3 + b^-4
>
>93 = b^4 + 2b^3 + b^2 + b^1 + b^-1 + b^-2 + 2b^-3 + b^-4
>
>94 = 2b^4 + 2b^-4
>
>95 = 2b^4 + 1 + 2b^-4
>
>96 = 2b^4 + 2 + 2b^-4
>
>97 = 2b^4 + b^1 + b^-1 + 2b^-4
>
>98 = 2b^4 + b^1 + 1 + b^-1 + 2b^-4
>
>99 = 2b^4 + b^1 + 2 + b^-1 + 2b^-4
>
>100 = 2b^4 + 2b^1 + 2b^-1 + 2b^-4
>
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