OFFSET
1,2
COMMENTS
For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y].
What is this sequence, asymptotically?
LINKS
Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Geometric Mean.
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))].
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan Vos Post, Feb 19 2006
EXTENSIONS
Corrected by R. J. Mathar, Jun 23 2014
Typo in a(40) corrected by Seth A. Troisi, May 13 2022
STATUS
approved