1,2

Eric Weisstein's World of Mathematics, Root-Mean-Square.

Eric Weisstein's World of Mathematics, Mean.

a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + ceiling(RMS[a(n),a(n-1)]). a(n+1) = a(n) + ceiling[Sqrt[[a(n)^2]+[a(n-1)^2]/2]].

It can easily be proved via induction that a(n)<=2^n. On the other hand we can derive a lower bound: We derive another sequence of the form b(n) = a*c^n, where "a" and "c" are real numbers. If b(1)<=a(1) and b(2)<=a(2) and a(n+1) = a(n)+Ceiling(Sqrt((a(n)^2+a(n-1)^2)/2)) >= b(n)+Sqrt((b(n)^2+b(n-1)^2)/2) >= b(n+1) then, via induction we can safely conclude that a(n)>=b(n). With this method we can derive that a(n) >= 1.80805^(n-1) (where 1.80... is the positive solution of x^2 = x+Sqrt((x^2+1)/2)). Hence we have 1.80805 < a(n)^(1/n) < 2. Can these bounds be improved? - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 21 2006

a(3) = 2 + ceiling[sqrt[(1^2 + 2^2)/2]] = 2 + ceiling[Sqrt[5/2]] = 2 + 2 = 4.

a(4) = 4 + ceiling[sqrt[(2^2 + 4^2)/2]] = 4 + ceiling[Sqrt[20/2]] = 4 + 4 = 8.

a(5) = 8 + ceiling[sqrt[(4^2 + 8^2)/2]] = 8 + ceiling[Sqrt[80/2]] = 8 + 7 = 15.

a(6) = 15 + ceiling[sqrt[(8^2 + 15^2)/2]] = 15 + ceiling[Sqrt[289/2]] = 15 + 13 = 28.

a(7) = 28 + ceiling[sqrt[(15^2 + 28^2)/2]] = 28 + ceiling[Sqrt[1009/2]] = 28 + 23 = 51.

a(8) = 51 + ceiling[sqrt[(28^2 + 51^2)/2]] = 51 + ceiling[Sqrt[3385/2]] = 51 + 42 = 93.

a(9) = 93 + ceiling[sqrt[(51^2 + 93^2)/2]] = 93 + ceiling[Sqrt[11250/2]] = 93 + 75 = 168 [the 75 is an exact value].

a(10) = 168 + ceiling[sqrt[(93^2 + 168^2)/2]] = 168 + ceiling[Sqrt[36873/2]] = 168 + 136 = 304.

a(11) = 304 + ceiling[sqrt[(168^2 + 304^2)/2]] = 304 + ceiling[Sqrt[120640/2]] = 304 + 246 = 550.

a(12) = 550 + ceiling[sqrt[(304^2 + 550^2)/2]] = 550 + ceiling[Sqrt[394916/2]] = 550 + 445 = 995.

Cf. A065094, A065095.

Sequence in context: A006727 A029907 A005682 this_sequence A065617 A062065 A008936

Adjacent sequences: A114830 A114831 A114832 this_sequence A114834 A114835 A114836

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 19 2006