[seqfan] Interesting set of palindromes
Dale Gerdemann
dale.gerdemann at googlemail.com
Wed Oct 19 15:47:13 CEST 2011
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
More information about the SeqFan
mailing list