# [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

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