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A189970
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Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).
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9
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2, 3, 1, 6, 5, 1, 2, 4, 2, 9, 1, 7, 3, 1, 3, 2, 3, 3, 0, 4, 5, 1, 6, 1, 3, 2, 1, 1, 6, 1, 7, 8, 2, 3, 3, 7, 6, 2, 4, 5, 7, 9, 3, 7, 3, 8, 5, 8, 1, 3, 8, 7, 0, 8, 1, 8, 9, 4, 0, 6, 4, 3, 0, 5, 4, 4, 0, 2, 7, 5, 9, 2, 1, 4, 3, 8, 5, 9, 8, 8, 7, 1, 3, 3, 7, 3, 0, 9, 4, 5, 7, 6, 8, 2, 5, 5, 4, 8, 1, 5, 4, 7, 2, 0, 1, 4, 5, 2, 5, 1, 1, 1, 5, 3, 5, 2, 6, 9, 8, 2
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OFFSET
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1,1
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COMMENTS
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Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10x))/4, where x=sqrt(5)). This rectangle can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,6,3,...] at A189971. For details, see A188635.
Decimal expansion of sqrt(r + r*sqrt(r + r*sqrt(r + ...))), where r = (1 + sqrt(5))/2 = golden ratio. - Ilya Gutkovskiy, Aug 24 2015
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LINKS
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EXAMPLE
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2.31651242917313233045161321161782337624579...
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MATHEMATICA
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r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, {r, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A189971 *)
RealDigits[N[%%, 120]] (* A189970 *)
N[%%%, 40]
RealDigits[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4, 10, 120][[1]] (* Harvey P. Dale, Sep 24 2015 *)
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PROG
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(PARI) default(realprecision, 1000); x=sqrt(5); (1+x+sqrt(14+10*x))/4 \\ Anders Hellström, Aug 24 2015
(Magma) (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4; // G. C. Greubel, Jan 12 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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