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A188725
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Decimal expansion of shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).
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3
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6, 4, 3, 8, 5, 0, 0, 9, 6, 3, 0, 6, 5, 4, 0, 8, 3, 9, 7, 2, 2, 3, 2, 3, 2, 5, 6, 3, 5, 9, 4, 6, 9, 1, 7, 2, 9, 2, 6, 2, 1, 6, 6, 5, 4, 0, 8, 1, 3, 2, 6, 1, 5, 2, 5, 6, 1, 0, 6, 5, 1, 7, 3, 2, 5, 8, 9, 5, 9, 2, 1, 2, 6, 3, 3, 4, 3, 7, 5, 1, 1, 6, 9, 3, 8, 6, 9, 6, 6, 9, 2, 7, 7, 2, 1, 5, 3, 0, 9, 8, 5, 0, 0, 3, 9, 3, 0, 2, 8, 1, 2, 1, 5, 8, 5, 8, 7, 0, 2, 3, 1, 6, 7, 6, 5, 3, 0, 9, 1, 5
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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 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + 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 (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., 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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6.4385009630654083972232325635946917292621665408132615256106...
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MAPLE
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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]] (* A188725 *)
ContinuedFraction[t, 120] (* A188726 *)
RealDigits[Pi + Sqrt[1 + Pi^2], 10, 100][[1]] (* G. C. Greubel, Oct 31 2018 *)
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PROG
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(PARI) default(realprecision, 100); Pi + sqrt(1 + Pi^2) \\ G. C. Greubel, Oct 31 2018
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R) + Sqrt(1 + Pi(R)^2); // G. C. Greubel, Oct 31 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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