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A095903
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Lexical ordering of the lazy Fibonacci representations.
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4
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1, 2, 3, 4, 5, 7, 6, 8, 9, 12, 10, 13, 15, 20, 11, 14, 16, 21, 17, 22, 25, 33, 18, 23, 26, 34, 28, 36, 41, 54, 19, 24, 27, 35, 29, 37, 42, 55, 30, 38, 43, 56, 46, 59, 67, 88, 31, 39, 44, 57, 47, 60, 68, 89, 49, 62, 70, 91, 75, 96, 109, 143, 32, 40, 45, 58, 48, 61, 69, 90, 50, 63
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OFFSET
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1,2
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COMMENTS
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A permutation of the natural numbers. As suggested by the example, the numbers can be generated as a graph consisting of two components, each being a tree. One tree has root 1 and consists of the numbers in the lower Wythoff sequence, A000201; the other has root 2 and consists of the numbers in the upper Wythoff sequence, A001950. (One could start with 0 and have a single tree instead of two components.)
Regard generations g(n) of the graph as rows of an array (see Example); then |g(n)| = 2^n. Every row includes exactly two Fibonacci numbers; specifically, row n includes F(2n) and F(2n+1). - Clark Kimberling, Mar 11 2015
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LINKS
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EXAMPLE
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Start with 1,2. Suffix the next two Fibonacci numbers, getting 1+2, 1+3; 2+3, 2+5. Suffix the next two Fibonacci numbers, getting 1+2+3, 1+2+5, 1+3+5, 1+3+8; 2+3+5, 2+3+8, 2+5+8, 2+5+13. Continue, obtaining
row 1: 1,2
row 2: 3,4,5,7
row 3: 6,8,9,12,10,13,15,20
row 4: 11,14,16,21,17,22,25,33,18,23,26,34,28,36,41,54
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MATHEMATICA
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Map[Total, Fibonacci[Flatten[NestList[Flatten[Map[{Join[#, {Last[#]+1}], Join[#, {Last[#]+2}]}&, #], 1]&, {{2}, {3}}, 7], 1]]] (* Peter J. C. Moses, Mar 06 2015 *)
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PROG
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(PARI) a(n) = n++; my(x=0, y=0); for(i=0, logint(n, 2)-1, y++; [x, y]=[y, x+y]; if(bittest(n, i), [x, y]=[y, x+y])); y; \\ Kevin Ryde, Jun 19 2021
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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