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A188883
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Decimal expansion of (1 + sqrt(1 + Pi^2))/Pi.
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1
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1, 3, 6, 7, 7, 4, 8, 3, 9, 4, 9, 3, 1, 3, 6, 7, 4, 4, 4, 6, 9, 9, 6, 9, 1, 7, 6, 5, 6, 8, 2, 2, 0, 5, 4, 5, 5, 6, 5, 1, 1, 1, 3, 2, 6, 8, 9, 0, 2, 1, 4, 8, 8, 6, 9, 4, 7, 5, 0, 0, 4, 6, 5, 7, 5, 6, 7, 1, 5, 3, 4, 5, 6, 2, 8, 2, 0, 1, 7, 6, 9, 3, 0, 7, 9, 0, 1, 9, 3, 0, 9, 7, 4, 1, 9, 3, 2, 3, 3, 5, 3, 1, 2, 2, 6, 6, 3, 0, 2, 7, 3, 4, 3, 3, 0, 8, 1, 4, 5, 9, 8, 2, 2, 8, 1, 5, 8, 9, 1, 9
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OFFSET
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1,2
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COMMENTS
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Decimal expansion of the length/width ratio of a (2/Pi)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (2/Pi)-extension rectangle matches the continued fraction [1,2,1,2,1,1,3,1,1,5,1,7,1,1,23,2,...] for the shape L/W = (1 + sqrt(1 + Pi^2))/Pi. 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 (2/Pi)-extension rectangle, 1 square is removed first, then 2 squares, then 1 square, then 2 squares, ..., so that the original rectangle of shape (1 + sqrt(1 + Pi^2))/Pi is partitioned into an infinite collection of squares.
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LINKS
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EXAMPLE
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1.36774839493136744469969176568220545565111326890...
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MATHEMATICA
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r = 2/Pi; 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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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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