A180657 Numerator of the fraction tan( Sum_{k=1..n} arctan(k) ).

1, -3, 0, 4, -9, 105, -308, 36, -423, 2387, -26004, 104472, -50617, 119889, -5466072, 3154072, 51692571, -2921193, 351666136, -1278405156, 11188330461, -68445012691, 553299094188, -4915961459556

Offset

1,2

Links

Table of n, a(n) for n=1..24.

T. Amdeberhan, Luis A. Medina and V. H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Number Theory 128 (2008) 1807-1846.

V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012. - From N. J. A. Sloane, Dec 22 2012

Formula

Let x(1)=1 and x(n) = (x(n-1)+n)/(1-n*x(n-1)). Then a(n) = numerator(x(n)).

Example

The fractions are x(1)=1, x(2)=-3, x(3)=0, x(4)=4, x(5)=-9/19, x(6)=105/73 etc.

Maple

A := proc(n) local x, itr; x := 1 ; for itr from 2 to n do x := (x+itr)/(1-itr*x) ; end do; numer(x) ; end proc:
seq(A(n), n=1..30) ;

Mathematica

x[1] := 1; x[n_] := (x[n - 1] + n)/(1 - n * x[n - 1]); Table[Numerator[x[n]], {n, 10}] (* Alonso del Arte, Jan 21 2011 *)
(* Just for verification: *)
x[n_] := Tan[Sum[ArcTan[k], {k, n}]] // TrigExpand;
Table[x[n] // Numerator, {n, 24}] (* Jean-François Alcover, Mar 29 2020 *)

Crossrefs

For denominators see A220447.

Sequence in context: A068627 A357884 A074171 * A094665 A309053 A052439

Adjacent sequences: A180654 A180655 A180656 * A180658 A180659 A180660

Keyword

sign,frac,easy

Author

R. J. Mathar, Jan 21 2011

Status

approved