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A225605 a(1) = least k such that 1/3 < H(k) - 1/3; a(2) = least k such that H(a(1)) - H(3) < H(k) - H(a(1)), and for n > 2, a(n) = least k such that H(a(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.

%I A225605 #10 Dec 25 2023 17:57:08
%S A225605 5,9,16,29,53,97,178,327,601,1105,2032,3737,6873,12641,23250,42763,
%T A225605 78653,144665,266080,489397,900141,1655617,3045154,5600911,10301681,
%U A225605 18947745,34850336,64099761,117897841,216847937,398845538,733591315,1349284789,2481721641
%N A225605 a(1) = least k such that 1/3 < H(k) - 1/3; a(2) = least k such that H(a(1)) - H(3) < H(k) - H(a(1)), and for n > 2, a(n) = least k such that H(a(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.
%C A225605 Suppose that x and y are positive integers and that x <=y.  Let a(1) = least k such that H(y) - H(x-1) < H(k) - H(y); let a(2) = least k such that H(a(1)) - H(y) < H(k) - H(a(1)); and for n > 2, let a(n) = least k such that greatest such H(a(n-1)) - H(a(n-2)) < H(k) - H(a(n-1)).  The increasing sequences H(a(n)) - H(a(n-1)) and a(n)/a(n-1) converge.  For what choices of (x,y) is the sequence a(n) linearly recurrent?
%C A225605 For A225605, (x,y) = (3,3); it appears that H(a(n)) - H(a(n-1)) approaches 0.60937786343... ; it is conjectured a(n)/a(n-1) approaches the constant given at A058265..
%H A225605 Clark Kimberling, <a href="/A225605/b225605.txt">Table of n, a(n) for n = 1..300</a>
%F A225605 a(n) = A192804(n+4)  (conjectured).
%F A225605 a(n) = 2*a(n-1) - a(n-4)  (conjectured).
%F A225605 G.f.: (5 - x - 2 x^2 - 3 x^3)/(1 - 2 x + x^4)  (conjectured)
%e A225605 The first two values (a(1),a(2)) = (5,9) match the beginning of the following inequality chain:
%e A225605   1/3 < 1/4 + 1/5 < 1/6 + 1/7 + 1/8 + 1/9 < ...
%t A225605 z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 3; y = 3;
%t A225605 a[1] = Ceiling[w /. FindRoot[h[w] == 2 h[y] - h[x - 1], {w, 1}, WorkingPrecision -> 400]]; a[2] = Ceiling[w /. FindRoot[h[w] == 2 h[a[1]] - h[y], {w, a[1]}, WorkingPrecision -> 400]]; Do[s = 0; a[t] = Ceiling[w /. FindRoot[h[w] == 2 h[a[t - 1]] - h[a[t - 2]], {w, a[t - 1]}, WorkingPrecision -> 400]], {t, 3, z}];
%t A225605 m = Map[a, Range[z]] (* A225605, _Peter J. C. Moses_, Jul 12 2013 *)
%Y A225605 Cf. A224820, A224965, A228016.
%K A225605 nonn,frac,easy
%O A225605 1,1
%A A225605 _Clark Kimberling_, Aug 03 2013