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A178255
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Decimal expansion of (3+sqrt(17))/2.
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6
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3, 5, 6, 1, 5, 5, 2, 8, 1, 2, 8, 0, 8, 8, 3, 0, 2, 7, 4, 9, 1, 0, 7, 0, 4, 9, 2, 7, 9, 8, 7, 0, 3, 8, 5, 1, 2, 5, 7, 3, 5, 9, 9, 6, 1, 2, 6, 8, 6, 8, 1, 0, 2, 1, 7, 1, 9, 9, 3, 1, 6, 7, 8, 6, 5, 4, 7, 4, 7, 7, 1, 7, 3, 1, 6, 8, 8, 1, 0, 7, 9, 6, 7, 9, 3, 9, 3, 1, 8, 2, 5, 4, 0, 5, 3, 4, 2, 1, 4, 8, 3, 4, 2, 2, 7
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OFFSET
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1,1
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COMMENTS
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Continued fraction expansion of (3+sqrt(17))/2 is A109007.
The rectangle R whose shape (i.e., length/width) is (3+sqrt(17))/2 can be partitioned into rectangles of shapes 3 and 3/2 in a manner that matches the periodic continued fraction [3, 3/2, 3, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [3, 1, 1, 3, 1, 1,...]. For details, see A188635. - Clark Kimberling, May 07 2011
The positive eigenvalue of the real symmetric 2 X 2 matrix M defined by M(i,j) = max(i,j) = [(1 2), (2 2)] is (3+sqrt(17))/2, while the negative one is (3-sqrt(17))/2. For a generalization, see A085984. - Bernard Schott, Apr 13 2020
The positive root of x^2 - 3^x - 2. The negative root is -(-3 + sqrt(17))/2 = -0.56155... - Wolfdieter Lang, Dec 10 2022
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LINKS
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EXAMPLE
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(3+sqrt(17))/2 = 3.56155281280883027491...
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MATHEMATICA
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FromContinuedFraction[{3, 3/2, {3, 3/2}}]
ContinuedFraction[%, 100] (* [3, 1, 1, 3, 1, 1, ...] *)
RealDigits[N[%%, 120]] (* A178255 *)
N[%%%, 40]
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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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