A228025 - OEIS

%I #8 Dec 25 2023 17:51:15

%S 20,76,285,1065,3976,14840,55385,206701,771420,2878980,10744501,

%T 40099025,149651600,558507376,2084377905,7779004245,29031639076,

%U 108347552060,404358569165,1509086724601,5631988329240,21018866592360,78443478040201,292755045568445

%N a(1) = least k such that 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 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 For A228025, (x,y) = (2,5); it appears that H(a(n)) - H(a(n-1)) approaches log(2 + sqrt(3)) and that and a(n)/a(n-1) approaches sqrt(3).

%H Clark Kimberling, <a href="/A228025/b228025.txt">Table of n, a(n) for n = 1..100</a>

%F a(n) = A061278(n+1)  (conjectured).

%F a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3)  (conjectured).

%F G.f.: (-20 + 24 x - 5 x^2)/(-1 + 5 x - 5 x^2 + x^3)  (conjectured)

%e The first two values (a(1),a(2)) = (20,76) match the beginning of the following inequality chain:  1/2+1/3+1/4+1/5 < 1/6+...+1/20 < 1/21+...+1/76 < ...

%t z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 2; y = 5;

%t 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 m = Map[a, Range[z]] (* A227653, _Peter J. C. Moses_, Jul 12 2013 *)

%Y Cf. A224820, A224965, A228016.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Aug 03 2013