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A190182
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Decimal expansion of (1+x+sqrt(8+2x))/4, where x=sqrt(15).
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2
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2, 2, 1, 0, 2, 7, 5, 5, 3, 2, 8, 1, 9, 0, 2, 0, 9, 6, 8, 7, 7, 8, 9, 7, 1, 3, 5, 2, 5, 0, 4, 8, 8, 7, 0, 5, 3, 3, 0, 4, 0, 8, 6, 3, 2, 9, 6, 7, 8, 3, 7, 4, 2, 9, 4, 7, 2, 8, 5, 6, 9, 4, 9, 7, 7, 4, 3, 9, 8, 4, 2, 5, 8, 6, 2, 0, 8, 9, 5, 9, 9, 2, 5, 0, 3, 7, 1, 1, 9, 9, 2, 9, 9, 8, 6, 7, 6, 0, 9, 2, 1, 4, 0, 3, 5, 9, 1, 3, 1, 1, 0, 6, 7, 8, 2, 5, 3, 3, 3, 8
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OFFSET
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1,1
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COMMENTS
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The rectangle R whose shape (i.e., length/width) is (1+x+sqrt(8+2x))/4, where x=sqrt(15), can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,1,r,1,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,4,1,3,10,...] at A190183. For details, see A188635.
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LINKS
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EXAMPLE
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2.210275532819020968778971352504887053304...
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MATHEMATICA
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r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (*A190183*)
N[%%%, 40]
RealDigits[(1 + Sqrt[15] + Sqrt[8 + 2*Sqrt[15]])/4, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)
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PROG
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(PARI) (1 + sqrt(15) + sqrt(8 + 2*sqrt(15)))/4) \\ G. C. Greubel, Dec 28 2017
(Magma) [(1 + Sqrt(15) + Sqrt(8 + 2*Sqrt(15)))/4]; // G. C. Greubel, Dec 28 2017
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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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