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A228016 a(1) = least k such that 1/1+1/2+1/3+1/4+1/5 < H(k) - H(5); a(2) = least k such that H(a(1)) - H(5) < 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 A228016 #7 Dec 25 2023 17:51:46
%S A228016 54,539,5340,52865,523314,5180279,51279480,507614525,5024865774,
%T A228016 49741043219,492385566420,4874114620985,48248760643434,
%U A228016 477613491813359,4727886157490160,46801248083088245,463284594673392294,4586044698650834699,45397162391834954700
%N A228016 a(1) = least k such that 1/1+1/2+1/3+1/4+1/5 < H(k) - H(5); a(2) = least k such that H(a(1)) - H(5) < 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 A228016 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 A228016 For A227965, (x,y) = (1,5); H(a(n)) - H(a(n-1)) approaches a limit 2.29243166956117768780078... and a(n)/a(n-1) approaches a limit 0.8989794855663561963945681494117
%H A228016 Clark Kimberling, <a href="/A228016/b228016.txt">Table of n, a(n) for n = 1..100</a>
%F A228016 a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3)  (conjectured).
%F A228016 G.f.: (-54 + 55 x - 5 x^2)/(-1 + 11 x - 11 x^2 + x^3)  (conjectured)
%e A228016 The first two values (a(1),a(2)) = (54,539) match the beginning of the following inequality chain:  1/1+1/2+1/3+1/4+1/5 < 1/6+...+1/54 < 1/55+...+1/539 < ...
%t A228016 z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 1; y = 5;
%t A228016 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 A228016 m = Map[a, Range[z]] (* A227653, _Peter J. C. Moses_, Jul 12 2013 *)
%Y A228016 Cf. A224820, A224965.
%K A228016 nonn,frac,easy
%O A228016 1,1
%A A228016 _Clark Kimberling_, Aug 02 2013