cp's OEIS Frontend

A227012 a(n) = floor(M(g(n-1)+1, ..., g(n))), where M = harmonic mean and g(n) = n^3.

%I A227012 #45 Nov 03 2024 04:33:23
%S A227012 1,4,16,43,91,166,275,422,614,857,1158,1521,1953,2460,3049,3724,4492,
%T A227012 5359,6332,7415,8615,9938,11391,12978,14706,16581,18610,20797,23149,
%U A227012 25672,28373,31256,34328,37595,41064,44739,48627,52734,57067,61630,66430,71473
%N A227012 a(n) = floor(M(g(n-1)+1, ..., g(n))), where M = harmonic mean and g(n) = n^3.
%C A227012 Suppose that f(k) is a sequence such that f(k) > 0 for k >= 1, the limit of f(k) is 0, and the sum of f(k) as k->oo diverges.  Let g(n) be a strictly increasing sequence of positive integers, and s(n) = Sum_{k=g(n-1)+1..g(n)} f(k). If f(k) = 1/k, then M(n) = (g(n) - g(n-1))/s(n) is the harmonic mean of g(n-1),...,g(n).
%C A227012 Conjecture: if f(k) = u/(v*k + w), where u,v,w are integers, and g(n) is a polynomial, then the sequence with n-th term m(n) = floor(M(n)) is linearly recurrent.  The conjecture extends to these cases, in which a,b,c,d are integers and a > 0:
%C A227012 (1) if g(n) = a*n^2 + b*n + c, the recurrence has order 2, and the first 3 recurrence coefficients for m(n) are 3, -3, 1; these are followed by some nonnegative number of 0's, a property abbreviated below as "(fbz)"; e.g., A002378.
%C A227012 (2) if g(n) has the form (a*n^2 + b*n + c)/2 where a and b are odd, then the recurrence has order 4, and the first 4 coefficients for m(n) are 2, 0-, -1, 2 (fbz); e.g., A080576.
%C A227012 (3) if g(n) = a*n^3 + b*n^2 + c*n + d, the recurrence has order 7, and the first 7 coefficients for m(n) are 3, -3, 1, 1, -3, 3, -1 (fbz); e.g., A227012.
%H A227012 Clark Kimberling, <a href="/A227012/b227012.txt">Table of n, a(n) for n = 1..1000</a>
%F A227012 a(n+2) = (1/8)*(27 - (-1)^n - 2*cos(n*Pi/2) + 2*sin(n*Pi/2) + 2*n*(4*n^2 + 18 n + 27)) for n >= 1 (conjectured).
%F A227012 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7) for n >= 10 (conjectured).
%F A227012 G.f.: x*(1 + x + 7*x^2 + 6*x^3 + 5*x^4 + 5*x^5 - 2*x^7 + x^8)/(((x - 1)^4)*(1 + x + x^2 + x^3)) (conjectured).
%F A227012 a(n) = (2*n^3 - 3*n^2 + n + 2)/2 + floor(max(0, n - 3)/4) (conjectured). - _Franck Maminirina Ramaharo_, Apr 12 2018
%e A227012 a(1) = floor(1/(1/1)) = 1, a(2) = floor(7/(1/2 + 1/3 + ... + 1/8)).
%t A227012 Clear[g]; g[n_] := g[n] = n^3; a = {1}; Do[AppendTo[a, Floor[(Last[#] - First[#] + 1)/(HarmonicNumber[Last[#]]-HarmonicNumber[First[#] - 1])] &[
%t A227012    N[{g[k - 1] + 1, g[k]}, 150]]], {k, 2, 100}]; a (* _Peter J. C. Moses_, Jul 03 2012 *)
%Y A227012 Cf. A227013, A227014, A227015, A227016, A227017.
%K A227012 nonn,easy
%O A227012 1,2
%A A227012 _Clark Kimberling_, Jul 01 2013