cp's OEIS Frontend

A198414 Decimal expansion of x > 0 satisfying x^2 = 2*sin(x).

%I A198414 #22 Jan 28 2025 16:12:38
%S A198414 1,4,0,4,4,1,4,8,2,4,0,9,2,4,3,4,3,6,4,1,4,8,3,2,7,9,4,3,7,4,5,7,5,8,
%T A198414 6,0,3,7,2,5,7,1,6,1,3,7,0,4,9,1,1,4,8,1,0,9,4,4,8,2,4,3,5,4,8,7,7,5,
%U A198414 2,5,2,9,5,6,1,7,1,4,4,3,6,2,1,2,0,5,1,0,1,5,2,4,8,2,0,8,1,7,5
%N A198414 Decimal expansion of x > 0 satisfying x^2 = 2*sin(x).
%C A198414 For many choices of a,b,c, there is a unique nonzero number x satisfying a*x^2+b*x=c*sin(x).
%C A198414 Specifically, for a>0 and many choices of b and c, the curves y=ax^2+bx and y=c*sin(x) meet in a single point if and only if b=c, in which case the curves have a common tangent line, y=c*x.  If b<c, the curves meet in quadrant 1; if b>c, they meet in quadrant 2.
%C A198414 Guide to related sequences (with graphs included in Mathematica programs):
%C A198414 a.....b.....c.....x
%C A198414 1.....0.....1.....A124597
%C A198414 1.....0.....2.....A198414
%C A198414 1.....0.....3.....A198415
%C A198414 1.....0.....4.....A198416
%C A198414 1.....1.....2.....A198417
%C A198414 1.....1.....3.....A198418
%C A198414 1.....1.....4.....A198419
%C A198414 1.....2.....1.....A198424
%C A198414 1.....2.....3.....A198425
%C A198414 1.....2.....4.....A198426
%C A198414 1....-1.....1.....A198420
%C A198414 1....-1.....1.....A198420
%C A198414 1....-1.....2.....A198421
%C A198414 1....-1.....3.....A198422
%C A198414 1....-2.....1.....A198427
%C A198414 1....-2.....2.....A198428
%C A198414 1....-2.....3.....A198429
%C A198414 1....-2.....4.....A198430
%C A198414 1....-3.....1.....A198431
%C A198414 1....-3.....2.....A198432
%C A198414 1....-3.....3.....A198433
%C A198414 1....-3.....4.....A198488
%C A198414 1....-4.....1.....A198489
%C A198414 1....-4.....2.....A198490
%C A198414 1....-4.....3.....A198491
%C A198414 1....-4.....4.....A198492
%C A198414 2.....0.....1.....A198583
%C A198414 2.....0.....3.....A198605
%C A198414 2.....1.....2.....A198493
%C A198414 2.....1.....3.....A198494
%C A198414 2.....1.....4.....A198495
%C A198414 2.....2.....1.....A198496
%C A198414 2.....2.....3.....A198497
%C A198414 2.....3.....1.....A198608
%C A198414 2.....3.....2.....A198609
%C A198414 2.....3.....4.....A198610
%C A198414 2.....4.....1.....A198611
%C A198414 2.....4.....3.....A198612
%C A198414 2....-1.....1.....A198546
%C A198414 2....-1.....2.....A198547
%C A198414 2....-1.....3.....A198548
%C A198414 2....-1.....4.....A198549
%C A198414 2....-2.....3.....A198559
%C A198414 2....-3.....1.....A198566
%C A198414 2....-3.....2.....A198567
%C A198414 2....-3.....3.....A198568
%C A198414 2....-3.....4.....A198569
%C A198414 2....-4.....1.....A198577
%C A198414 2....-4.....3.....A198578
%C A198414 3.....0.....1.....A198501
%C A198414 3.....0.....2.....A198502
%C A198414 3.....1.....2.....A198498
%C A198414 3.....1.....3.....A198499
%C A198414 3.....1.....4.....A198500
%C A198414 3.....2.....1.....A198613
%C A198414 3.....2.....3.....A198614
%C A198414 3.....2.....4.....A198615
%C A198414 3.....3.....1.....A198616
%C A198414 3.....3.....2.....A198617
%C A198414 3.....3.....4.....A198618
%C A198414 3.....4.....1.....A198606
%C A198414 3.....4.....2.....A198607
%C A198414 3.....4.....3.....A198619
%C A198414 3....-1.....1.....A198550
%C A198414 3....-1.....2.....A198551
%C A198414 3....-1.....3.....A198552
%C A198414 3....-1.....4.....A198553
%C A198414 3....-2.....1.....A198560
%C A198414 3....-2.....2.....A198561
%C A198414 3....-2.....3.....A198562
%C A198414 3....-2.....4.....A198563
%C A198414 3....-3.....1.....A198570
%C A198414 3....-3.....2.....A198571
%C A198414 3....-3.....4.....A198572
%C A198414 3....-4.....1.....A198579
%C A198414 3....-4.....2.....A198580
%C A198414 3....-4.....3.....A198581
%C A198414 3....-4.....4.....A198582
%C A198414 4.....0.....1.....A198503
%C A198414 4.....0.....3.....A198504
%C A198414 4.....1.....2.....A198505
%C A198414 4.....1.....3.....A198506
%C A198414 4.....1.....4.....A198507
%C A198414 4.....2.....1.....A198539
%C A198414 4.....2.....3.....A198540
%C A198414 4.....3.....1.....A198541
%C A198414 4.....3.....2.....A198542
%C A198414 4.....3.....4.....A198543
%C A198414 4.....4.....1.....A198544
%C A198414 4.....4.....3.....A198545
%C A198414 4....-1.....1.....A198554
%C A198414 4....-1.....2.....A198555
%C A198414 4....-1.....3.....A198556
%C A198414 4....-1.....4.....A198557
%C A198414 4....-1.....1.....A198554
%C A198414 4....-2.....1.....A198564
%C A198414 4....-2.....3.....A198565
%C A198414 4....-3.....1.....A198573
%C A198414 4....-3.....2.....A198574
%C A198414 4....-3.....3.....A198575
%C A198414 4....-3.....4.....A198576
%C A198414 Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
%C A198414 For an example related to A198414, take f(x,u,v)=x^2+u*x-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
%H A198414 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A198414 1.4044148240924343641483279437457586037...
%t A198414 (* Program 1: A198414 *)
%t A198414 a = 1; b = 0; c = 2;
%t A198414 f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x]
%t A198414 Plot[{f[x], g[x]}, {x, -1, 2}]
%t A198414 r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.41}, WorkingPrecision -> 110]
%t A198414 RealDigits[r] (* A198414 *)
%t A198414 (* Program 2: an implicit surface of x^2+u*x=v*sin(x) *)
%t A198414 f[{x_, u_, v_}] := x^2 + u*x - v*Sin[x];
%t A198414 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .01, 6}]}, {u, .1, 100}, {v, u, 100}];
%t A198414 ListPlot3D[Flatten[t, 1]]
%Y A198414 Cf. A197737.
%K A198414 nonn,cons
%O A198414 1,2
%A A198414 _Clark Kimberling_, Oct 24 2011
%E A198414 Edited by _Georg Fischer_, Aug 01 2021