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A188543
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Decimal expansion of the angle B in the doubly silver triangle ABC.
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6
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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
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=1+sqrt(2), the silver ratio.
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EXAMPLE
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B=0.4235466615478147887414222095779154 approximately.
B=24.2674 degrees approximately.
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MATHEMATICA
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r = 1+2^(1/2); Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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