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

1, 2, 3, 5, 8, 14, 24, 41, 71, 122, 211, 365, 632, 1094, 1895, 3282, 5684, 9845, 17052, 29534, 51154, 88601, 153461, 265802, 460382, 797405, 1381145, 2392213, 4143434, 7176638, 12430301, 21529913, 37290903, 64589738, 111872708, 193769214, 335618123, 581307641, 1006854369, 1743922922

Offset: 1

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• nonn
• easy

Comments

For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y].

What is this sequence, asymptotically?

Examples

			a(3) = 2 + floor(2*1*2/(1+2)) = 2 + floor(4/3) = 2 + 1 = 3.
a(4) = 3 + floor(2*2*3/(2+3)) = 3 + floor(12/5) = 3 + 2 = 5.
a(5) = 5 + floor(2*3*5/(3+5)) = 5 + floor(30/8) = 5 + 3 = 8.
a(6) = 8 + floor(2*5*8/(5+8)) = 8 + floor(80/13) = 8 + 6 = 14.
a(7) = 14 + floor(2*8*14/(8+14)) = 14 + floor(112/11) = 14 + 10 = 24.
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Mean.
• Eric Weisstein's World of Mathematics, Geometric Mean.

Crossrefs

Cf. A065094, A065095.

Programs

Maple

hMean := proc(a,b)
    2*a*b/(a+b) ;
end proc:
A114831 := proc(n)
    option remember;
    if n<= 2 then
        n;
    else
        procname(n-1)+floor(hMean(procname(n-1),procname(n-2))) ;
    end if;
end proc:
seq(A114831(n),n=1..60) ; # R. J. Mathar, Jun 23 2014

Formula

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))].

Extensions

Corrected by R. J. Mathar, Jun 23 2014

Typo in a(40) corrected by Seth A. Troisi, May 13 2022