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A114831 Each term is previous term plus floor of harmonic mean of two previous terms.

%I A114831 #21 Feb 16 2025 08:33:00
%S A114831 1,2,3,5,8,14,24,41,71,122,211,365,632,1094,1895,3282,5684,9845,17052,
%T A114831 29534,51154,88601,153461,265802,460382,797405,1381145,2392213,
%U A114831 4143434,7176638,12430301,21529913,37290903,64589738,111872708,193769214,335618123,581307641,1006854369,1743922922
%N A114831 Each term is previous term plus floor of harmonic mean of two previous terms.
%C A114831 For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y].
%C A114831 What is this sequence, asymptotically?
%H A114831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean.</a>
%H A114831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeometricMean.html">Geometric Mean.</a>
%F A114831 a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + floor(HarmonicMean[a(n),a(n-1)]). a(n+1) = a(n) + floor[(2*a(n)*a(n-1))/(a(n)+a(n-1))].
%e A114831 a(3) = 2 + floor(2*1*2/(1+2)) = 2 + floor(4/3) = 2 + 1 = 3.
%e A114831 a(4) = 3 + floor(2*2*3/(2+3)) = 3 + floor(12/5) = 3 + 2 = 5.
%e A114831 a(5) = 5 + floor(2*3*5/(3+5)) = 5 + floor(30/8) = 5 + 3 = 8.
%e A114831 a(6) = 8 + floor(2*5*8/(5+8)) = 8 + floor(80/13) = 8 + 6 = 14.
%e A114831 a(7) = 14 + floor(2*8*14/(8+14)) = 14 + floor(112/11) = 14 + 10 = 24.
%p A114831 hMean := proc(a,b)
%p A114831     2*a*b/(a+b) ;
%p A114831 end proc:
%p A114831 A114831 := proc(n)
%p A114831     option remember;
%p A114831     if n<= 2 then
%p A114831         n;
%p A114831     else
%p A114831         procname(n-1)+floor(hMean(procname(n-1),procname(n-2))) ;
%p A114831     end if;
%p A114831 end proc:
%p A114831 seq(A114831(n),n=1..60) ; # _R. J. Mathar_, Jun 23 2014
%Y A114831 Cf. A065094, A065095.
%K A114831 nonn,easy
%O A114831 1,2
%A A114831 _Jonathan Vos Post_, Feb 19 2006
%E A114831 Corrected by _R. J. Mathar_, Jun 23 2014
%E A114831 Typo in a(40) corrected by _Seth A. Troisi_, May 13 2022