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 A255240 Decimal expansion of 1/(2*cos(Pi/7)). 4
 5, 5, 4, 9, 5, 8, 1, 3, 2, 0, 8, 7, 3, 7, 1, 1, 9, 1, 4, 2, 2, 1, 9, 4, 8, 7, 1, 0, 0, 6, 4, 1, 0, 4, 8, 1, 0, 6, 7, 2, 8, 8, 8, 6, 2, 4, 7, 0, 9, 1, 0, 0, 8, 9, 3, 7, 6, 0, 2, 5, 9, 6, 8, 2, 0, 5, 1, 5, 7, 5, 3, 5, 9, 4, 2, 9, 0, 5, 3, 6, 1, 8, 5, 0, 8, 3, 7, 8, 9, 4, 7, 8, 3, 8, 5, 4, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This is the decimal expansion of t = 1/rho(7) = 2 + rho(7) - rho(7)^2 with rho(7) = 2*cos(Pi/7) the length ratio of the smaller diagonal and the side of a regular heptagon. See A160389 for the decimal expansion of rho(7). t satisfies the cubic equation t^3 - 2*t^2 - t + 1 = 0. t = 1/rho(7) is the slope tan(alpha) appearing in Archimedes's neusis construction of the regular heptagon. The corresponding angle alpha is approximately 29,028 degrees. See the link, Figure 1, also for references. From Peter Bala, Oct 16 2021: (Start) t = sin(Pi/7)/sin(2*Pi/7). The other roots of the cubic equation t^3 - 2*t^2 - t + 1 = 0 are t_1 = 1/(1 - t) = sin(3*Pi/7)/sin(6*Pi/7) = 2.2469796037... and t_2 = 1/(1 - t_1) = - sin(2*Pi/7)/sin(4*Pi/7) = - 0.8019377358.... Compare with A231187 and A160389. The algebraic number field Q(t) is a totally real cubic field of discriminant 7^2 and class number 1 with a cyclic Galois group over Q of order 3.  See Shanks. (End) LINKS Wolfdieter Lang, Archimedes's Construction of the Regular Heptagon. D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152 FORMULA 1/rho(7) = 1/(2*cos(Pi/7)) = 0.55495813208... From Peter Bala, Oct 10 2021: (Start) t = 2*(cos(Pi/7) - cos(2*Pi/7)); t_1 = 2*(cos(3*Pi/7) - cos(6*Pi/7)); t_2 = 2*(cos(5*Pi/7) - cos(10*Pi/7)). t = Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+2)*(7*n+5)) = 1 - Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+3)*(7*n+4)) =  1 - A255241. (End) MATHEMATICA RealDigits[1/(2*Cos[Pi/7]), 10, 100][] (* Georg Fischer, Apr 04 2020 *) CROSSREFS Cf. A116425, A160389, A231187, A255241. Sequence in context: A011189 A261346 A011409 * A168578 A344235 A019253 Adjacent sequences:  A255237 A255238 A255239 * A255241 A255242 A255243 KEYWORD nonn,cons,changed AUTHOR Wolfdieter Lang, Mar 12 2015 EXTENSIONS Name corrected by Georg Fischer, Apr 04 2020 STATUS approved

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Last modified October 28 04:43 EDT 2021. Contains 348313 sequences. (Running on oeis4.)