editing

approved

a(1) = least k such that 1/2 + 1/3 < H(k) - H(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.

approved

editing

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

proposed

approved

editing

proposed

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}];

m = Map[a, Range[z]] (* A227653 , Peter Moses, Jul 12 2013 *)

Cf. A224820, A224865A224965.

For A227965, (x,y) = (2,3); H(a(n)) - H(a(n-1)) approaches a limit 0.9348448455...by A227966, , and a(n)/a(n-1) approaches a limit 2.546818276...

allocated a(1) = least k such that 1/2 + 1/3 < H(k) - H(3); a(2) = least k such that H(a(1)) - H(3) < H(k) -H(a(1)), and for Clark Kimberlingn > 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.

8, 21, 54, 138, 352, 897, 2285, 5820, 14823, 37752, 96148, 244872, 623645, 1588311, 4045140, 10302237, 26237926, 66823230, 170186624, 433434405, 1103878665, 2811378360, 7160069791, 18235396608, 46442241368, 118279949136, 301237536249, 767197263003

1,1

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?

For A227965, (x,y) = (2,3); H(a(n)) - H(a(n-1)) approaches a limit 0.9348448455...by A227966, and a(n)/a(n-1) approaches a limit 2.546818276...

Clark Kimberling, <a href="/A227653/b227653.txt">Table of n, a(n) for n = 1..100</a>

a(n) = A077849(n+1) (conjectured).

a(n) = 3*a(n-1) - a(n-2) - a(n-4) (conjectured).G.f.: (8 - 3 x - x^2 - 3 x^3)/(1 - 3 x + x^2 + x^4) (conjectured).

The first two values (a(1),a(2)) = (8,21) match the beginning of the following inequality chain: 1/2 + 1/3 < 1/4 + ... + 1/8 < 1/9 + ... + 1/21 < ...

z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 2; y = 3;

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}];

m = Map[a, Range[z]] (* A227653 *)

Cf. A224820, A224865.

allocated

nonn,frac,easy

Clark Kimberling, Aug 02 2013

approved

editing

allocated for Clark Kimberling

recycled

allocated

editing

approved