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A227014 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n^5.

%I A227014 #17 Nov 03 2024 04:54:07
%S A227014 1,10,104,543,1883,5102,11717,23906,44626,77735,128110,201769,305989,
%T A227014 449428,642243,896212,1224852,1643541,2169636,2822595,3624095,4598154,
%U A227014 5771249,7172438,8833478,10788947,13076362,15736301,18812521,22352080
%N A227014 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n^5.
%C A227014 See A227012.  It is conjectured that A227014 is a linear recurrence sequence with signature (5,-10,10,-5,1,...Z...,1,-5,-10,-10,-1,0,0), where ...Z... represents a string of 138 zeros; has been confirmed for a(1), a(2),..., a(150000).
%H A227014 Clark Kimberling, <a href="/A227014/b227014.txt">Table of n, a(n) for n = 1..1000</a>
%e A227014 a(1) = floor(1/(1/1)) = 1.
%e A227014 a(2) = floor(31/(1/2 + 1/3 + ... + 1/32)) = 10.
%t A227014 Clear[g]; g[n_] := N[n^5, 100]; a = {1}; Do[AppendTo[a, Floor[(#2 - #1 + 1)/(HarmonicNumber[#2]-HarmonicNumber[#1 - 1])] &[g[k - 1] + 1, g[k]]], {k, 2, 200}]; a (* _Peter J. C. Moses_, Jul 05 2012 *)
%t A227014 (* confirm generating function *)
%t A227014 p = {1, -4, 5, 9, 54, 117, 117, 122, 118, 122, 118, 122, 118, 122,
%t A227014    118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
%t A227014    122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
%t A227014    118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
%t A227014    122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
%t A227014    119, 117, 129, 107, 134, 106, 134, 106, 134, 106, 134, 106, 134,
%t A227014    106, 134, 106, 134, 107, 129, 117, 119, 122, 118, 122, 118, 122,
%t A227014    118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
%t A227014    122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
%t A227014    118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
%t A227014    122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
%t A227014    117, 126, 113, 113, 64, 5, 1};
%t A227014 q = {0, 0, 1, -5, 10, -10, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%t A227014    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 5, -10, 10, -5,
%t A227014     1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[Factor[gf], {x, 0, 100}], x] (* _Peter J. C. Moses_, Jul 08 2012 *)
%Y A227014 Cf. A227012, A227013.
%K A227014 nonn,easy
%O A227014 1,2
%A A227014 _Clark Kimberling_, Jul 01 2013