|
| |
|
|
A316713
|
|
Unique representation of nonnegative numbers by iterated tribonacci A, B and C sequences.
|
|
5
|
|
|
|
1, 21, 121, 31, 1121, 221, 131, 11121, 2121, 1221, 321, 1131, 231, 111121, 21121, 12121, 3121, 11221, 2221, 1321, 11131, 2131, 1231, 331, 1111121, 211121, 121121, 31121, 112121, 22121, 13121, 111221, 21221, 12221, 3221, 11321, 2321, 111131, 21131, 12131, 3131, 11231, 2231, 1331, 11111121, 2111121, 1211121, 311121, 1121121, 221121, 131121, 1112121, 212121, 122121, 32121, 113121, 23121, 1111221, 211221, 121221, 31221, 112221, 22221, 13221, 111321, 21321, 12321
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
This representation is the tribonacci A000073 analog of the Wythoff representation of numbers (A189921 or A317208) for the Fibonacci case.
The complementary and disjoint sets A, B and C are given by the sequences A278040, A278039, and A278041, respectively.
The present representation uses 1 for B, 2 for A and 3 for C numbers. The brackets for sequence iteration and the final argument 0 have to be added. E.g.: a(0) = 1 for B(1), a(1) = 21 for A(B(0)), a(2) = 121 for B(A(B(0))), a(3) = 31 for C(B(0)), ...
The length of the string a(n) is A316714(n). The number of B, A and C sequences used for the ABC-representation of n (that is the number of 1s, 2s and 3s of a(n)) is A316715, A316716 and A316717, respectively.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The complementary and disjoint sequences A, B, C begin, for n >= 0:
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
A: 1 5 8 12 14 18 21 25 29 32 36 38 42 45 49 52 56 58 62 65 69 73 76 ...
B: 0 2 4 6 7 9 11 13 15 17 19 20 22 24 26 28 30 31 33 35 37 39 41 ...
C: 3 10 16 23 27 34 40 47 54 60 67 71 78 84 91 97 104 108 115 121 128 135 141 ...
---------------------------------------------------------------------------------
The ABC representations begin:
#(1) #(2) #(3) L(a(n))
n = 0: 1 B(0) = 0 1 0 0 1
n = 1: 21 A(B(0)) = 1 1 1 0 2
n = 2: 121 B(A(B(0))) = 2 2 1 0 3
n = 3: 31 C(B(0)) = 3 1 0 1 2
n = 4: 1121 B(B(A(B(0)))) = 4 3 1 0 4
n = 5: 221 A(A(B(0))) = 5 1 2 0 3
n = 6: 131 B(C(B(0))) = 6 2 0 1 3
n = 7: 11121 B(B(B(A(B(0))))) = 7 4 1 0 5
n = 8: 2121 A(B(A(B(0)))) = 8 2 2 0 4
n = 9: 1221 B(A(A(B(0)))) = 9 2 2 0 4
n = 10: 321 C(A(B(0))) = 10 1 1 1 3
n = 11: 1131 B(B(C(B(0)))) = 11 3 0 1 4
n = 12: 231 A(C(B(0))) = 12 1 1 1 3
n = 13: 111121 B(B(B(B(A(B(0)))))) = 13 5 1 0 6
n = 14: 21121 A(B(B(A(B(0))))) = 14 3 2 0 5
n = 15: 12121 B(A(B(A(B(0))))) = 15 3 2 0 5
n = 16: 3121 C(B(A(B(0)))) = 16 2 1 1 4
n = 17: 11221 B(B(A(A(B(0))))) = 17 3 2 0 5
n = 18: 2221 A(A(A(B(0)))) = 18 1 3 0 4
n = 19: 1321 B(C(A(B(0)))) = 19 2 1 1 4
n = 20: 11131 B(B(B(C(B(0))))) = 20 4 0 1 5
...
----------------------------------------------------------------------------
|
|
|
CROSSREFS
|
Cf. A000073, A003144, A003145, A003146, A189921, A317208, A278040, A278039, A278041, A316714, A316715, A316716, A316717, A317208.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
