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A189642 Numerator of H(n+4) - H(n), where H(n) = Sum_{k=1..n} 1/k.

%I A189642 #30 Oct 20 2024 01:40:51
%S A189642 25,77,19,319,533,275,1207,1691,763,3013,3875,543,6061,7409,2981,
%T A189642 10675,12617,4927,17179,19823,2525,25897,29351,11033,37153,41525,
%U A189642 15409,51271,56669,6937,68575,75107,27347,89389,97163,35125,114037,123161,14751,142843,153425
%N A189642 Numerator of H(n+4) - H(n), where H(n) = Sum_{k=1..n} 1/k.
%C A189642 a(n) = Numerator of (4*n^3+30*n^2+70*n+50)/((n+1)*(n+2)*(n+3)*(n+4)).
%C A189642 (4*n^3+30*n^2+70*n+50)/a(n) has period length 9, repeating 1, 1, 9, 1, 1, 3, 1, 1, 3.
%C A189642 It is of interest to note that the roots of 4*n^3+30*n^2+70*n+50 are -phi-2, phi-3, and -5/2, where phi = (1+sqrt(5))/2.
%C A189642 H(n+4)-H(n) = (2*n^3+15*n^2+35*n+25)/(12*binomial(n+4,4)).
%H A189642 T. D. Noe, <a href="/A189642/b189642.txt">Table of n, a(n) for n = 0..1000</a>
%F A189642 a(n) = (4*n^3+30*n^2+70*n+50)/(2*(1+2*P(3, n+1))*(1+2*P(9, n+7))), where P(k, n) = floor(1/2*cos(2*n*Pi/k)+3/5).
%p A189642 h:=n-> sum(1/k,k=1..n):seq(numer(h(n+4)-h(n)), n=0..30);
%p A189642 # alternative Maple program:
%p A189642 P:=(k,n)-> floor(1/2*cos(2*n*Pi/k)+3/5):
%p A189642 seq((4*n^3+30*n^2+70*n+50)/(2*(1+2*P(3,n+1))*(1+2*P(9,n+7))),n=0..30);
%t A189642 Numerator[Table[HarmonicNumber[n+4] - HarmonicNumber[n], {n, 0, 100}]] (* _T. D. Noe_, May 24 2011 *)
%o A189642 (Magma) Harmonic:=func< n | n eq 0 select 0 else &+[ 1/k: k in [1..n] ] >; A189642:=func< n | Numerator( Harmonic(n+4)-Harmonic(n) ) >; [ Numerator( A189642(n) ): n in [0..40] ]; // _Klaus Brockhaus_, May 21 2011
%Y A189642 Cf. A001008, A002805.
%K A189642 nonn,easy
%O A189642 0,1
%A A189642 _Gary Detlefs_, May 02 2011