

A188543


Decimal expansion of the angle B in the doubly silver triangle ABC.


6



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

0,1


COMMENTS

There is a unique (shape of) triangle ABC that is both sidesilver and anglesilver. Its angles are B, t*B and piBt*B, where t is the silver ratio, 1+sqrt(2), at A014176. "Sidesilver" and "anglesilver" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,2,2,...] of t. For doubly golden and doubly eratio triangles, see A152149 and A188544. For the side partitioning and angle partitioning (i,e, constructions in which 2 triangles are removed at each stage, analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle into squares) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.


LINKS

Table of n, a(n) for n=0..119.
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165171.


FORMULA

B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=1+sqrt(2), the silver ratio.


EXAMPLE

B=0.4235466615478147887414222095779154 approximately.
B=24.2674 degrees approximately.


MATHEMATICA

r = 1+2^(1/2); Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision > 120][[1, 2]]][[1]]


CROSSREFS

Cf. A152149, A014176, A188544.
Sequence in context: A200024 A247206 A228047 * A182272 A182273 A166016
Adjacent sequences: A188540 A188541 A188542 * A188544 A188545 A188546


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Apr 03 2011


STATUS

approved



