A188544 Decimal expansion of the angle B in the doubly e-ratio triangle ABC.

3, 6, 8, 9, 3, 1, 2, 7, 4, 9, 4, 7, 8, 0, 5, 8, 4, 2, 6, 5, 1, 9, 1, 1, 2, 7, 2, 6, 8, 8, 6, 4, 0, 8, 5, 7, 1, 8, 6, 8, 3, 4, 4, 2, 8, 8, 3, 5, 2, 6, 1, 9, 0, 6, 5, 9, 8, 5, 4, 6, 2, 1, 2, 1, 1, 1, 1, 7, 6, 5, 9, 8, 7, 5, 6, 8, 4, 9, 0, 6, 0, 6, 7, 0, 1, 2, 1, 0, 6, 0, 4, 8, 5, 9, 8, 4, 2, 8, 4, 2, 8, 0, 9, 9, 1, 1, 8, 2, 1, 8, 8, 5, 9, 6, 9, 8, 4, 4, 2, 2, 9, 4, 8, 7, 3, 4, 6, 8

Offset

0,1

Comments

There is a unique (shape of) triangle ABC that is both side-e-ratio and angle-e-ratio. Its angles are B, t*B and pi-B-t*B, where t=e. "Side-e-ratio" and "angle-e-ratio" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,1,2,1,1,4,1,1,6,...] of t. For doubly golden and doubly silver triangles, see A152149 and A188543. For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

Links

Table of n, a(n) for n=0..127.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Formula

B is the number in [0,Pi] such that sin(B*e^2)=e*sin(B).

Example

B=0.36893127494780584265191127268864 approximately.
B=21.1382 degrees approximately.

Mathematica

r = E; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

Crossrefs

Cf. A152149, A188543.

Sequence in context: A023993 A350618 A133159 * A163463 A137386 A358510

Adjacent sequences: A188541 A188542 A188543 * A188545 A188546 A188547

Keyword

nonn,cons

Author

Clark Kimberling, Apr 03 2011

Extensions

a(127) corrected by Sean A. Irvine, Sep 08 2021

Status

approved