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A226271
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Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.
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3
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1, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142
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OFFSET
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1,2
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COMMENTS
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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).
Apart from initial terms, the same as A001611=(1, 2, 2, 3, 4, 6,...), A020706=(4,6,9,...), A048577=(3, 4, 6, ...), A000381=(2, 3, 4, ...).
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LINKS
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FORMULA
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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
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EXAMPLE
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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,...
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MATHEMATICA
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LinearRecurrence[{2, 0, -1}, {1, 4, 6, 9}, 40] (* Harvey P. Dale, Feb 04 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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