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A079936 Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5).

%I A079936 #20 Mar 25 2024 06:37:45
%S A079936 1,2,5,13,17,34,305,610,1597,4181,5473,10946,98209,196418,514229,
%T A079936 1346269,1762289,3524578,31622993,63245986,165580141,433494437,
%U A079936 567451585,1134903170,10182505537,20365011074,53316291173,139583862445
%N A079936 Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5).
%C A079936 The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.
%H A079936 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,322,0,0,0,0,0,-1).
%F A079936 For n>=0, a(6n+1)=A001076(4n+1); a(6n+2)=2a(6n+1); a(6n+3)=A001076(4n+1)+A001076(4n+2); a(6n+4)=A001076(4n+3)-A001076(4n+2); a(6n+5)=A001076(4n+3); a(6n+6)=2a(6n+5). Asymptotics: a(6n) -> 2*sqrt(5)*(tau)^(12n-3); a(6n+2)/a(6n+1) -> (tau)^2; a(6n+3)/a(6n+2) -> (tau)^2; a(6n+4)/a(6n+3) -> (tau)^2/2; a(6n+6)/a(6n+5) -> (tau)^6/2; where tau = (1+sqrt(5))/2.
%F A079936 G.f.: -x*(x -1)*(2*x^10 +3*x^9 +8*x^8 +21*x^7 +55*x^6 +72*x^5 +38*x^4 +21*x^3 +8*x^2 +3*x +1) / (x^12 -322*x^6 +1). - _Colin Barker_, Jun 16 2013
%e A079936 a(4) = 13 since frac(1x) + frac(2x) + frac(5x) + frac(13x) < 1, while frac(1x) + frac(2x) + frac(5x) + frac(k*x) > 1 for all k>5 and k<13.
%Y A079936 Cf. A001076 (denominators of convergents to sqrt(5)), A079934, A079935, A079937.
%K A079936 nonn,easy
%O A079936 1,2
%A A079936 _Benoit Cloitre_ and _Paul D. Hanna_, Jan 21 2003