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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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

There is a unique (shape of) triangle ABC that is both side-silver and angle-silver. Its angles are B, t*B and pi-B-t*B, where t is the silver ratio, 1+sqrt(2), at A014176. "Side-silver" and "angle-silver" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,2,2,...] of t.  For doubly golden and doubly e-ratio 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) 165-171.

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

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Last modified August 21 05:40 EDT 2019. Contains 326162 sequences. (Running on oeis4.)