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A079934 Greedy frac multiples of sqrt(2): a(1)=1, Sum_{n>=0} frac(a(n)*x)=1 at x=sqrt(2).

%I A079934 #36 Mar 25 2024 06:37:35
%S A079934 1,3,5,10,17,29,46,99,169,268,577,985,1562,3363,5741,9104,19601,33461,
%T A079934 53062,114243,195025,309268,665857,1136689,1802546,3880899,6625109,
%U A079934 10506008,22619537,38613965,61233502,131836323,225058681,356895004,768398401,1311738121
%N A079934 Greedy frac multiples of sqrt(2): a(1)=1, Sum_{n>=0} frac(a(n)*x)=1 at x=sqrt(2).
%C A079934 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 A079934 G. C. Greubel, <a href="/A079934/b079934.txt">Table of n, a(n) for n = 1..1000</a>
%H A079934 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A079934 For n > 0, a(3*n) = A000129(2*n+1).
%F A079934 a(3*n+2) = a(3*n) + A000129(2*n+2).
%F A079934 a(3*n+4) = a(3*n+2) + a(3*n+3).
%F A079934 a(3*n) = ceiling((3+2*sqrt(2))^n*(2+sqrt(2))/4).
%F A079934 a(3*n+2)/a(3*n+1) -> 1/sqrt(2).
%F A079934 a(3*n+1)/a(3*n) -> 3-sqrt(2).
%F A079934 a(3*n)/a(3*n-1) -> (8+5*sqrt(2))/7.
%F A079934 G.f.: x*(2*x^9 - 13*x^6 - x^5 - x^4 + 4*x^3 + 5*x^2 + 3*x + 1) / (x^6 - 6*x^3 + 1). - _Colin Barker_, Jun 16 2013
%e A079934 a(4) = 10 since frac(1x) + frac(3x) + frac(5x) + frac(10x) < 1, while frac(1x) + frac(3x) + frac(5x) + frac(k*x) > 1 for all k > 5 and k < 10.
%t A079934 CoefficientList[Series[(1 + 3*z + 5*z^2 + 4*z^3 - z^4 - z^5 - 13*z^6 + 2*z^9)/(1 - 6*z^3 + z^6), {z, 0, 40}], z] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2012 *)
%o A079934 (PARI) x='x+O('x^50); Vec(x*(2*x^9 -13*x^6 -x^5 -x^4 +4*x^3 +5*x^2 +3*x +1)/(x^6-6*x^3 +1)) \\ _G. C. Greubel_, Sep 22 2017
%Y A079934 Cf. A000129 (Pell numbers), A078343, A079935, A079936.
%K A079934 nonn,easy
%O A079934 1,2
%A A079934 _Benoit Cloitre_ and _Paul D. Hanna_, Jan 20 2003