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A188737
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Decimal expansion of (7+sqrt(85))/6.
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3
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2, 7, 0, 3, 2, 5, 7, 4, 0, 9, 5, 4, 8, 8, 1, 4, 5, 5, 1, 6, 6, 7, 0, 4, 5, 7, 1, 3, 6, 2, 7, 1, 3, 2, 1, 9, 2, 8, 7, 4, 4, 6, 7, 5, 0, 8, 1, 2, 0, 4, 1, 0, 6, 6, 8, 0, 0, 1, 2, 9, 2, 0, 3, 4, 2, 4, 0, 4, 4, 5, 1, 7, 1, 1, 3, 3, 6, 4, 5, 9, 1, 0, 1, 2, 7, 9, 8, 2, 3, 4, 8, 4, 6, 5, 5, 4, 6, 7, 6, 0, 8, 2, 3, 3, 8, 9, 9, 6, 8, 1, 4, 6, 4, 7, 8, 6, 1, 4, 0, 2, 5, 3, 5, 4, 1, 1, 0, 5, 5, 7
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OFFSET
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1,1
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COMMENTS
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Decimal expansion of the length/width ratio of a (7/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (7/3)-extension rectangle matches the continued fraction [2,1,2,2,1,2,2,1,2,2,1,...] for the shape L/W=(7+sqrt(85))/6. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the (7/3)-extension rectangle, 2 squares are removed first, then 1 square, then 2 squares, then 2 squares,..., so that the original rectangle of shape (7+sqrt(85))/6 is partitioned into an infinite collection of squares.
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LINKS
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EXAMPLE
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2.703257409548814551667045713627132192874467508120...
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MAPLE
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MATHEMATICA
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r = 7/3; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
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PROG
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(PARI) default(realprecision, 100); (7+sqrt(85))/6 \\ G. C. Greubel, Nov 01 2018
(Magma) SetDefaultRealField(RealField(100)); (7+Sqrt(85))/6; // G. C. Greubel, Nov 01 2018
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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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