A263920 A positive integer n is in this sequence iff arctan(n)^2 can be represented as Sum_{0<k<n} c(k)*arctan(k)^2 with rational c(k). The terms are in increasing order.

7, 47, 57, 99, 117

Offset

1,1

Comments

The terms given are certainly in the sequence. Although I lack a rigorous proof that no intermediate terms were omitted, an extensive computer search gave no other candidates in between.

It is an open question if the sequence is infinite.

Links

Table of n, a(n) for n=1..5.

Eric Weisstein's MathWorld, Inverse Tangent.

Example

7 is in the sequence, because arctan(7)^2 = -5*arctan(1)^2 + (10/3)*arctan(2)^2 + (2/3)*arctan(3)^2.
47 is in the sequence, because arctan(47)^2 = (2939/210)*arctan(2)^2 - (125/21)*arctan(3)^2 - (6/5)*arctan(4)^2 - (12/7)*arctan(5)^2 - (29/7)*arctan(7)^2 + (15/7)*arctan(8)^2 + (2/5)*arctan(13)^2 + (11/7)*arctan(18)^2 - arctan(21)^2 + (7/10)*arctan(38)^2.

Crossrefs

Cf. A005528, A002312.

Sequence in context: A089725 A086040 A009241 * A124837 A122731 A059452

Adjacent sequences: A263917 A263918 A263919 * A263921 A263922 A263923

Keyword

nonn,hard,more

Author

Vladimir Reshetnikov, Oct 29 2015

Status

approved