A226271 Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.
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
Offset: 1
Examples
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,...
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
Programs
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Mathematica
LinearRecurrence[{2,0,-1},{1,4,6,9},40] (* Harvey P. Dale, Feb 04 2016 *) -
PARI
A226271(n)=if(n>1,fibonacci(n+2))+1
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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
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
Comments