OFFSET
1,2
COMMENTS
The Fibonacci ordering of the rationals (cf. A226080) is the sequence of rationals produced from the initial vector [1] by appending iteratively the new rationals obtained by applying the map t-> (t+1, 1/t) to the vector (cf. example).
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
a(n) = 2*a(n-1)-a(n-3) for n>4. G.f.: -x*(2*x^3+2*x^2-2*x-1) / ((x-1)*(x^2+x-1)). - Colin Barker, Jun 03 2013
a(n) = 1+(2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5))+(1+sqrt(5))^n*(3+sqrt(5))))/sqrt(5) for n>1. - Colin Barker, May 11 2016
E.g.f.: -2*(1 + x) + exp(x) + (3*sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, May 11 2016
EXAMPLE
Starting from the vector [1] and applying the map t->(1+t,1/t), we get [2,1] (but ignore the number 1 which already occurred earlier), then [3,1/2], then [4,1/3,3/2,2] (where we ignore 2), etc. This yields the sequence (1,2,3,1/2,4,1/3,3/2,5,1/4,4/3,5/2,2/3,....) The unit fractions 1=1/1, 1/2, 1/3, ... occur at positions 1,4,6,9,...
MATHEMATICA
LinearRecurrence[{2, 0, -1}, {1, 4, 6, 9}, 40] (* Harvey P. Dale, Feb 04 2016 *)
PROG
(PARI) A226271(n)=if(n>1, fibonacci(n+2))+1
(PARI) {k=1; print1(s=1, ", "); U=Set(g=[1]); for(n=1, 9, U=setunion(U, Set(g=select(f->!setsearch(U, f), concat(apply(t->[t+1, k/t], g))))); for(i=1, #g, numerator(g[i])==1&&print1(s+i", ")); s+=#g)} \\ for illustrative purpose
(PARI) Vec(-x*(2*x^3+2*x^2-2*x-1)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, May 11 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, Jun 01 2013
STATUS
approved