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A188724
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Decimal expansion of shape of a (Pi/2)-extension rectangle; shape = (1/4)*(Pi + sqrt(16 + Pi^2)).
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1
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2, 0, 5, 6, 9, 5, 2, 4, 3, 8, 7, 1, 0, 9, 6, 5, 9, 0, 9, 3, 9, 6, 7, 8, 7, 9, 2, 4, 3, 7, 8, 8, 0, 7, 2, 5, 8, 5, 8, 8, 0, 9, 9, 1, 4, 1, 5, 4, 9, 7, 1, 7, 6, 2, 0, 4, 6, 7, 6, 4, 2, 6, 8, 3, 4, 1, 6, 1, 9, 5, 6, 5, 7, 6, 0, 3, 4, 1, 7, 4, 6, 1, 3, 2, 2, 1, 8, 2, 6, 6, 1, 4, 5, 7, 6, 5, 0, 2, 1, 5, 1, 8, 9, 6, 9, 9, 2, 5, 3, 9, 6, 2, 4, 2, 1, 0, 6, 6, 2, 4, 8, 0, 9, 8, 2, 4, 8, 8, 4, 1, 9, 8
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OFFSET
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1,1
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COMMENTS
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See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.
A (Pi/2)-extension rectangle matches the continued fraction [2,17,1,1,3,1,3,2,2,1637,1,210,7,...] of the shape L/W = (1/4)*(Pi + sqrt(16 + Pi^2)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (Pi/2)-extension rectangle, 2 squares are removed first, then 17 squares, then 1 square, then 1 square, then 3 squares, ..., so that the original rectangle is partitioned into an infinite collection of squares.
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LINKS
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EXAMPLE
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2.0569524387109659093967879243788072585880991...
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MATHEMATICA
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r = Pi/2; 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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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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