|
%I #41 Jan 06 2023 15:24:04
%S 4,2,0,3,7,2,3,3,9,4,2,3,2,2,3,0,7,5,6,4,0,9,9,3,0,0,6,6,4,6,2,2,1,8,
%T 7,3,9,4,9,1,8,9,8,6,6,6,0,0,6,1,1,8,7,1,2,9,1,6,5,4,6,6,4,6,8,6,5,5,
%U 3,3,7,0,8,8,5,9,7,9,0,8,0,3,5,5,7,4,3,9,0,5,6,0,3,9,2,8,3,3,6
%N Decimal expansion of one of four fixed points (mod 1) of Minkowski's question mark function.
%C Other fixed points (mod 1) are 0, 1/2 and 1-A048819. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 10 2006
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.9 Minkowski-Bower constant, pp. 441-443.
%H Jean-François Alcover, <a href="/A048819/a048819_2.gif">Graph of the question mark function.</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010208143502/http://www.mathsoft.com/asolve/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [From the Wayback machine]
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Minkowski-BowerConstant.html">Minkowski-Bower Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html">Minkowski's Question Mark Function</a>
%H <a href="/index/Me#MinkowskiQ">Index entries for sequences related to Minkowski's question mark function</a>
%e 0.4203723394232230756409930066462218739491898666...
%t digits = 99; n0 = 3; dx = 10^-n0; qm[x_] := (ac = Accumulate[ContinuedFraction[x, 200]]; 2 + 2*Sum[(-1)^n* 2^(-ac[[n]]), {n, 1, Length[ac]}]); x = dx; While[N[qm[x], digits+5] < x, x = x + dx]; x0 = x - dx; Do[dx = 10^-n; x = x0; While[N[qm[x], digits+5] < x, x = N[x + dx, digits+5]]; x0 = x - dx , {n, n0+1, digits}]; RealDigits[x0, 10, digits] // First (* _Jean-François Alcover_, Oct 13 2014 *)
%t RealDigits[x /. FindRoot[MinkowskiQuestionMark[x] - x, {x, .42, .421}, WorkingPrecision -> 200, MaxIterations -> 500], 10, 99][[1]] (* _Eric W. Weisstein_, Jan 06 2023 *)
%Y Cf. A048817-A048822.
%K nonn,cons
%O 0,1
%A _Christian G. Bower_, Apr 15 1999
|
