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Search: id:A152149
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| A152149 |
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Decimal expansion of the angle B in the triangle ABC that is both side-golden and angle-golden. |
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+0
1
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| 6, 5, 7, 4, 0, 5, 4, 8, 2, 9, 7, 6, 5, 3, 2, 5, 9, 2, 3, 8, 0, 9, 6, 8, 5, 4, 1, 5, 2, 9, 3, 9, 7, 1, 2, 6, 5, 4, 1, 4, 9, 5, 9, 4, 6, 4, 8, 7, 8, 3, 9, 3, 7, 0, 7, 8, 2, 0, 9, 2, 8, 0, 8, 5, 8, 8, 5, 3, 9, 5, 0, 6, 1, 3, 8, 1, 7, 7, 3, 5, 0, 7, 0, 1, 7, 1, 5, 1, 6, 5, 4, 4, 0, 5, 2, 2, 7, 8, 0, 5, 2, 8, 1, 2, 6
(list; cons; graph; listen)
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OFFSET
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0,1
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COMMENT
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There is a unique (shape of) triangle ABC that is both side-golden and
angle-golden. Its angles are B, t*B and pi-B-t*B, where t is the golden
ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC,
each in a manner that matches the continued fraction [1,1,1,...] of t.
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REFERENCES
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Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions," Journal for Geometry and Graphics, 11 (2007) 165-171.
Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.
Jordi Dou, Clark Kimberling and Laurence Kuipers, "A Fibonacci sequence of nested triangles," Problem S29, Amer. Math. Monthly 89 (1982) 696-697; proposed 87 (1980) 302.
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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+5^(1/2))/2, the golden ratio.
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EXAMPLE
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The number B begins with 0.65740548 (equivalent to 37.666559... degrees)
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CROSSREFS
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Cf. A000045.
Sequence in context: A101634 A071176 A089826 this_sequence A086268 A021156 A063046
Adjacent sequences: A152146 A152147 A152148 this_sequence A152150 A152151 A152152
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KEYWORD
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nonn,cons
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Nov 26 2008
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EXTENSIONS
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Keyword:cons added and offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 18 2009
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