|
%I #14 Dec 25 2023 17:52:14
%S 11,53,249,1164,5435,25371,118428,552798,2580343,12044484,56221045,
%T 262427666,1224955522,5717827134,26689578960,124581175389,
%U 581517950673,2714399875409,12670230858892,59141894115145,276061555506087,1288595564424512,6014885070144844
%N a(1) = least k such that 1 + 1/2 < H(k) - H(2); a(2) = least k such that H(a(1)) - 1/2 < 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 A227965, (x,y) = (1,2); H(a(n)) - H(a(n-1)) approaches a limit 1.540684... given by A227966, and a(n)/a(n-1) approaches a limit 4.6677834... given by A227967. It is unknown whether the sequence a(n) is linearly recurrent.
%H Clark Kimberling, <a href="/A227965/b227965.txt">Table of n, a(n) for n = 1..100</a>
%e The first two values (a(1),a(2)) = (11,53) match the beginning of the following inequality chain (and partition of the harmonic numbers): 1/1 + 1/2 < 1/3 + ... + 1/11 < 1/12 + ... + 1/53 < ...
%t z = 300; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 1; y = 2;
%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]] (* A227965 *)
%t t = N[Table[h[a[t]] - h[a[t - 1]], {t, 2, z, 25}], 60]
%t Last[RealDigits[t, 10]] (* A227966 *)
%t t = N[Table[a[t]/a[t - 1], {t, 2, z, 50}], 60]
%t Last[RealDigits[t, 10]] (* A227967 *)
%t (* A227965, _Peter J. C. Moses_, Jul 12 2013*)
%Y Cf. A224820, A224868, A227728, A227966, A227967.
%K nonn,frac,easy
%O 1,1
%A _Clark Kimberling_, Aug 01 2013
|
