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 A197682 Decimal expansion of Pi/(2 + 2*Pi). 32
 3, 7, 9, 2, 7, 3, 4, 9, 6, 4, 9, 7, 3, 8, 8, 0, 7, 2, 6, 7, 2, 2, 1, 5, 3, 4, 4, 5, 2, 2, 4, 4, 6, 4, 3, 2, 0, 6, 9, 2, 1, 3, 1, 8, 2, 8, 2, 0, 2, 6, 5, 4, 9, 8, 3, 3, 4, 4, 9, 4, 1, 0, 6, 8, 9, 1, 2, 7, 4, 0, 6, 8, 5, 5, 0, 4, 7, 8, 6, 8, 8, 1, 6, 0, 3, 1, 6, 5, 8, 7, 0, 0, 7, 6, 7, 7, 8, 8, 6 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x). If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x). Related sequences, each with a Mathematica program which includes a graph: ... b.....c.......sequence........x 1.....2.......A019673........ x = Pi/6 1.....3.......A019678........ x = Pi/8 1.....4.......(A000796)/10... x = Pi/10 1.....Pi......A197682........ x = Pi/(2+2*Pi) 1.....2*Pi....A197683........ x = Pi/(2+4*Pi) 1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi) 1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi) 1.....Pi/2....A197686........ x = Pi/(2+Pi) 1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi) 1.....Pi/4....A197688........ x = 2*Pi/(4+Pi) 1.....Pi/6....A197689........ x = 3*Pi/(6+Pi) 2.....3.......(A000796)/10... x = Pi/10 2.....Pi......A197690........ x = Pi/(4+2*Pi) 2.....2*Pi....A197691........ x = Pi/(4+4*Pi) 2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi) 2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi) 2.....Pi/2....A197694........ x = Pi/(4+Pi) 3.....Pi......A197695........ x = Pi/(2+2*Pi) 3.....2*Pi....A197696........ x = Pi/(6+4*Pi) 3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi) 3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi) 3.....Pi/2....A197699........ x = Pi/(6+Pi) 1/2...Pi......A197700........ x = Pi/(1+2*Pi) 1/2...2*Pi....A197701........ x = Pi/(1+4*Pi) 1/2...1/Pi....A197724........ x = Pi^2/(2+Pi) 1/2...2/Pi....A197725........ x = Pi^2/(4+Pi) 1/2...Pi/2....A197726........ x = Pi/(1+Pi) 1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi) 1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi) 1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi) 2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi) 1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi) 1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi) 1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi) 1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi) 1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi) 1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi) LINKS Table of n, a(n) for n=0..98. Index entries for transcendental numbers EXAMPLE 0.37927349649738807267221534452244643... MATHEMATICA b = 1; c = Pi; t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}] N[Pi/(2*b + 2*c), 110] RealDigits[%] (* A197682 *) Simplify[Pi/(2*b + 2*c)] Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}] PROG (PARI) 1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022 CROSSREFS Cf. A197683. Sequence in context: A131712 A072845 A197481 * A021729 A198236 A071641 Adjacent sequences: A197679 A197680 A197681 * A197683 A197684 A197685 KEYWORD nonn,cons AUTHOR Clark Kimberling, Oct 17 2011 STATUS approved

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Last modified June 3 13:22 EDT 2023. Contains 363110 sequences. (Running on oeis4.)