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Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

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`%I #7 Feb 11 2020 04:23:26
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`%S 6,7,5,1,7,7,4,4,2,4,5,8,5,5,7,1,3,9,8,1,3,2,8,5,6,2,5,0,7,5,8,2,7,6,
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`%T 3,3,6,8,4,0,7,3,1,5,8,9,8,9,0,5,1,5,9,1,4,0,1,1,3,0,8,1,0,8,5,9,1,5,
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`%U 1,9,7,2,9,4,9,8,4,0,2,3,7,9,3,5,3,3,7
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`%N Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.
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`%C Consider the "add the square root" sequence {b(n)} starting with 1, i.e., b(1)=1, b(n+1) = b(n) + sqrt(b(n)) for n > 0: at each step, the next term is obtained simply by adding the current term to its square root, so the sequence begins {1, 2, 2 + sqrt(2), 2 + sqrt(2) + sqrt(2 + sqrt(2)), ...}, i.e., {1, 2, 3.414213..., 5.261972..., ...}.
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`%C The 1st, 10th, 100th, and 1000th terms are
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`%C b(10^0) = 1.0
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`%C b(10^1) = 25.956...
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`%C b(10^2) = 2452.850...
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`%C b(10^3) = 248949.869...
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`%C b(10^4) = 24983729.385...
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`%C b(10^5) = 2499779700.562...
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`%C Since the difference between successive terms is b(n+1) - b(n) = sqrt(b(n)) and the similar differential equation (d/dx)f(x) = sqrt(f(x)) is satisified by the function f(x) = (1/4)*x^2, it is perhaps not surprising that lim_{n->infinity} b(n)/n^2 = 1/4. More precisely, it can be shown that, as n increases, b(n) approaches
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`%C (1/4)*n^2
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`%C + u*n
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`%C + u^2 - u/2
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`%C + (-(1/2)*u^2 + u/4 - 1/96)/n
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`%C + ((1/3)*u^3 + 0*u^2 + (-5/48)*u + 7/576)/n^2
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`%C + ...
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`%C where u = -(1/4)*log(n) + c
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`%C and c = 0.675177442458557139813285625075...
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`%C It follows that lim_{n->infinity} (b(n)/n - (n - log(n))/4) = c.
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`%C If we were to define {b(n)} instead as b(1)=1, b(n+1) = b(n) + 1/b(n) for n > 0 (i.e., "add the reciprocal" rather than "add the square root"), we would obtain the rational-valued sequence {b(n)} = {1, 2, 5/2, 29/10, 941/290, ...} (see A073833 and, for a constant arising from that sequence, A233770).
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`%e 0.67517744245855713981328562507582763368407315898905...
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`%Y Cf. A073833, A233770.
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`%K nonn,cons
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`%O 0,1
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`%A _Jon E. Schoenfield_, Feb 10 2020
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