A220447 -id:A220447 - cp's OEIS frontend

A220449 Define u(n) as in A220448; then a(1)=1, thereafter a(n) = u(n)*a(n-1).

1, 1, -10, 10, 190, -730, -6620, 55900, 365300, -5864300, -28269800, 839594600, 2691559000, -159300557000, -238131478000, 38894192662000, -15194495654000, -11911522255750000, 29697351895900000, 4477959179352100000, -21683886333440500000, -2029107997508660900000, 15145164178973569000000

Offset: 1

N. J. A. Sloane, Dec 22 2012

sign

The reason for including this sequence as well as A105750 is that the values of this sequence modulo various primes are of interest (see Moll).

N. J. A. Sloane, Table of n, a(n) for n = 1..100
V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012. See f_n.

Cf. A105750, A220446, A220447, A220448, A220450, A220451.

x:=proc(n) option remember;
if n=1 then 1 else (x(n-1)+n)/(1-n*x(n-1)); fi; end;
f:=proc(n) option remember; global x;
if n = 1 then 1 else n*x(n-1)*f(n-1)-f(n-1); fi; end;
[seq(f(n),n=1..30)];

A105750(n) = (-1)^(n+1)*a(n).

Define x(n) as in A220447. Then x(n) = (a(n+1)+a(n))/((n+1)*a(n)).

A220448 Define a sequence u(n) by u(1)=1; thereafter u(n) = f(n)/f(n-1) where f(n) = (-1)^(n+1)*A105750(n); sequence gives numerator(u(n)).

Original entry on oeis.org

1, 1, -10, -1, 19, -73, 662, -2795, 281, -4511, 21746, -322921, 1035215, -720817, 1077518, -87995911, -34376687, 929280875, -92673902, 6986985769, -33833494045, 243540693677, -1817775985570, 13097400661955, -27199287430171, 8249498995171439, -112427012362483262, 3008079144099400625, -10365435567354511181

Offset: 1

N. J. A. Sloane, Dec 22 2012

Also u(n) = n*x(n-1)-1, where x(n) is defined in A220447.

			The sequence u(n) begins 1, 1, -10, -1, 19, -73/19, 662/73, -2795/331, 281/43, -4511/281, 21746/4511, ...

V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012.

Denominator(u(n)) = A220447(n-1).

A220448 := proc(n)
    if n= 1 then
        1 ;
    else
        -A105750(n)/A105750(n-1) ;
        numer(%) ;
    end if;
end proc: # R. J. Mathar, Jan 04 2013

x[n_] := x[n] = If[n == 1, 1, (x[n-1] + n)/(1 - n*x[n-1])];
u[n_] := If[n == 1, 1, n*x[n-1] - 1];
a[n_] := Numerator[u[n]];
Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Aug 09 2023 *)

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

Original entry on oeis.org

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

R. J. Mathar, Jan 21 2011

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

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

For denominators see A220447.

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) ;

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 *)

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

cp's OEIS Frontend

A220449 Define u(n) as in A220448; then a(1)=1, thereafter a(n) = u(n)*a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A220448 Define a sequence u(n) by u(1)=1; thereafter u(n) = f(n)/f(n-1) where f(n) = (-1)^(n+1)*A105750(n); sequence gives numerator(u(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula