A095903 - OEIS

%I #18 Jun 19 2021 12:41:07

%S 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,

%T 28,36,41,54,19,24,27,35,29,37,42,55,30,38,43,56,46,59,67,88,31,39,44,

%U 57,47,60,68,89,49,62,70,91,75,96,109,143,32,40,45,58,48,61,69,90,50,63

%N Lexical ordering of the lazy Fibonacci representations.

%C 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.)

%C 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

%H Clark Kimberling, <a href="/A095903/b095903.txt">Table of n, a(n) for n = 1..4000</a>

%e 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

%e row 1:  1,2

%e row 2:  3,4,5,7

%e row 3:  6,8,9,12,10,13,15,20

%e row 4:  11,14,16,21,17,22,25,33,18,23,26,34,28,36,41,54

%t 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 *)

%o (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

%Y Cf. A000045, A095791, A000201, A001950, A255773 (the 1-tree), A255774 (the 2-tree).

%K easy,nonn

%O 1,2

%A _Clark Kimberling_, Jun 12 2004