A249022 - OEIS

%I #9 Oct 27 2014 20:50:52

%S 4,6,6,5,9,9,3,0,6,2,0,3,7,2,9,2,6,5,2,2,1,7,3,4,2,2,0

%N Decimal expansion of Sine Euler constant.

%C The Sine Euler constant is introduced here as the limit as n increases without bound of sum{sin(1/k), k = 1..n} - integral{sin(1/x) over [1,n]}; this is analogous to the Euler constant, defined as the limit of sum{1/k, k = 1..n} - integral{1/x over [1,n]}.

%e Sine Euler constant = 0.466599306203729265221734220...

%t f = DifferenceRoot[Function[{\[FormalY], \[FormalN]}, {((2 \[FormalN] - z) (2 \[FormalN] - (z + 1))) \[FormalY][\[FormalN]] + \[FormalY][1 + \[FormalN]] == 0, \[FormalY][1] == -1}]];

%t (Total[Table[1/((-1)^(n + 1) (2 n - 1)!) HarmonicNumber[k, 2 n - 1], {n, 50}]] /. k -> #) - (CosIntegral[1] - CosIntegral[1/#] - Sin[1] + # Sin[1/#]) &[N[10^35, 40]]

%t RealDigits[t][[1]]

%t (* _Peter J. C. Moses_, Oct 20 2014 *)

%Y Cf. A001620 (Euler constant), A249023 (Tangent Euler constant).

%K nonn,easy,cons

%O 0,1

%A _Clark Kimberling_, Oct 22 2014