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A188882
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Decimal expansion of (5+sqrt(34))/3.
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0
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3, 6, 1, 0, 3, 1, 7, 2, 9, 8, 2, 8, 1, 7, 6, 6, 8, 2, 3, 6, 2, 4, 7, 1, 7, 6, 2, 5, 8, 4, 8, 5, 2, 7, 6, 9, 2, 1, 7, 3, 7, 9, 9, 4, 4, 4, 9, 6, 1, 9, 9, 0, 6, 5, 1, 4, 8, 3, 3, 3, 5, 5, 8, 1, 6, 2, 2, 6, 0, 3, 3, 5, 3, 9, 9, 8, 9, 0, 4, 2, 0, 9, 2, 2, 1, 7, 4, 6, 7, 7, 5, 4, 8, 4, 3, 4, 5, 1, 3, 2, 8, 5, 2, 2, 6, 3, 2, 0, 7, 3, 5, 8, 4, 5, 1, 6, 3, 7, 1, 1, 7, 2, 7, 1, 2, 9, 1, 2, 0, 5
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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 (10/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (10/3)-extension rectangle matches the continued fraction [3,1,1,1,1,3,3,1,1,1,1,3,3,1,1,1,1,3,3,...] for the shape L/W=(5+sqrt(34))/3. 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 (10/3)-extension rectangle, 3 squares are removed first, then 1 square, then 1 square, then 1 square,..., so that the original rectangle of shape (5+sqrt(34))/3 is partitioned into an infinite collection of squares.
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LINKS
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EXAMPLE
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3.61031729828176682362471762584852769217379944...
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MATHEMATICA
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r = 10/3; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[(5+Sqrt[34])/3, 10, 140][[1]] (* Harvey P. Dale, Feb 18 2015 *)
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PROG
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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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