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A090458
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Decimal expansion of solution to n/x = x-n for n = 3.
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4
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3, 7, 9, 1, 2, 8, 7, 8, 4, 7, 4, 7, 7, 9, 2, 0, 0, 0, 3, 2, 9, 4, 0, 2, 3, 5, 9, 6, 8, 6, 4, 0, 0, 4, 2, 4, 4, 4, 9, 2, 2, 2, 8, 2, 8, 8, 3, 8, 3, 9, 8, 5, 9, 5, 1, 3, 0, 3, 6, 2, 1, 0, 6, 1, 9, 5, 3, 4, 3, 4, 2, 1, 2, 7, 7, 3, 8, 8, 5, 4, 4, 3, 3, 0, 2, 1, 8, 0, 7, 7, 9, 7, 4, 6, 7, 2, 2, 5, 1, 6, 3
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1...]. (See the Mathematica program below.) [From Clark Kimberling, May 05 2011]
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FORMULA
| n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
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MATHEMATICA
| FromContinuedFraction[{3, 1, {3, 1}}]
ContinuedFraction[%, 20]
RealDigits[N[%%, 120]] (*A090458*)
N[%%%, 40]
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PROG
| (PARI) solve(x=3, 4, x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011
(PARI) (3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011
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CROSSREFS
| Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Sequence in context: A177689 A191611 A101366 * A131712 A072845 A197481
Adjacent sequences: A090455 A090456 A090457 * A090459 A090460 A090461
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KEYWORD
| easy,nonn,cons
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AUTHOR
| Felix Tubiana (fat2(AT)columbia.edu), Feb 05 2004
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EXTENSIONS
| Additional comments from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 02 2004
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