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A114833
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Each term is previous term plus ceiling of root mean square of two previous terms.
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1
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0, 1, 2, 4, 8, 15, 28, 51, 93, 168, 304, 550, 995, 1799, 3253, 5882, 10635, 19229, 34767, 62861, 113656, 205497, 371550, 671782, 1214618, 2196094, 3970654, 7179153, 12980288, 23469047, 42433278, 76721609, 138716724, 250807167, 453472612, 819902445, 1482426947, 2680306255, 4846135343
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OFFSET
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0,3
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LINKS
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Eric Weisstein's World of Mathematics, Mean.
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FORMULA
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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, Feb 21 2006
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EXAMPLE
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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.
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MATHEMATICA
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a[n_] := a[n] = a[n - 1] + Ceiling[ Sqrt[(a[n - 1]^2 + a[n - 2]^2)/2]]; a[0] = 0; a[1] = 1; Array[a, 39, 0] (* Robert G. Wilson v, Jun 22 2014 *)
nxt[{a_, b_}]:={b, b+Ceiling[Sqrt[(a^2+b^2)/2]]}; Transpose[NestList[nxt, {0, 1}, 40]][[1]] (* Harvey P. Dale, May 12 2015 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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