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%I #6 Feb 07 2019 08:55:42
%S 5,6,1,5,4,9,0,9,6,8,7,2,6,9,9,1,3,1,0,8,4,8,7,4,7,1,4,0,8,6,2,7,6,2,
%T 5,8,5,9,9,1,1,3,4,3,5,7,1,6,5,0,9,5,5,6,3,4,2,3,3,8,4,5,5,0,8,0,2,9,
%U 4,5,0,8,6,1,1,8,3,8,3,6,5,6,9,6,8,2,2
%N Decimal expansion of lim_{k->infinity} (-k - (1/6)*log(k) + Sum_{j=1..k} sqrt(j)*arcsin(1/sqrt(j))).
%C Consider the curve lying between the positive x-axis and the line y=1 and generated by the following process for k = 1, 2, 3, ...: Draw a circular arc about the origin from (sqrt(k-1), 1) down to (sqrt(k), 0), then draw a vertical line segment from there up to (sqrt(k), 1).
%C After the k-th arc and line segment have been drawn, the length of the curve is k + Sum_{j=1..k} sqrt(j)*arcsin(1/sqrt(j)) = 2k + (1/6)*log(k) + C + (1/120)/k + (13/10080)/k^2 - (37/120960)/k^3 - (617/3548160)/k^4 + (8719/98841600)/k^5 + (47623/553512960)/k^6 - ... where C = 0.561549... is the constant whose decimal expansion consists of the terms of this sequence.
%e 0.56154909687269913108487471408627625859911343571650...
%Y Cf. A323736, A323737.
%K nonn,cons
%O 0,1
%A _Jon E. Schoenfield_, Feb 07 2019
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