A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.
1, 6, 9, 5, 6, 2, 0, 7, 6, 9, 5, 5, 9, 8, 6, 2, 0, 5, 7, 4, 1, 6, 3, 6, 7, 1, 0, 0, 1, 1, 7, 5, 3, 5, 3, 4, 2, 6, 1, 8, 1, 7, 9, 3, 8, 8, 2, 0, 8, 5, 0, 7, 7, 3, 0, 2, 2, 1, 8, 7, 0, 7, 2, 8, 4, 4, 5, 2, 4, 4, 5, 3, 4, 5, 4, 0, 8, 0, 0, 7, 2, 2, 1, 3, 9, 9
Offset: 1
Examples
1.6956207695598620574163671001175353426181793882085077...
References
- D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.
Links
- Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
- Index entries for algebraic numbers, degree 3.
Crossrefs
Programs
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Mathematica
z = 2000; r = 8/5; u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x]; (* A289260 *) v = N[u[[z]]/u[[z - 1]], 200] RealDigits[v, 10][[1]] (* A289265 *) -
PARI
solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019
Formula
r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019