A323738 Decimal expansion of lim_{k->infinity} (-k - (1/6)*log(k) + Sum_{j=1..k} sqrt(j)*arcsin(1/sqrt(j))).

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, 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, 4, 5, 0, 8, 6, 1, 1, 8, 3, 8, 3, 6, 5, 6, 9, 6, 8, 2, 2

Offset

0,1

Comments

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).

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.

Links

Table of n, a(n) for n=0..86.

Example

0.56154909687269913108487471408627625859911343571650...

Crossrefs

Cf. A323736, A323737.

Sequence in context: A080130 A188640 A198419 * A222133 A198728 A345144

Adjacent sequences: A323735 A323736 A323737 * A323739 A323740 A323741

Keyword

nonn,cons

Author

Jon E. Schoenfield, Feb 07 2019

Status

approved