%I #20 Aug 01 2021 12:56:30
%S 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 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 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 Decimal expansion of x > 0 satisfying x^2 = 2*sin(x).
%C For many choices of a,b,c, there is a unique nonzero number x satisfying a*x^2+b*x=c*sin(x).
%C 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 Guide to related sequences (with graphs included in Mathematica programs):
%C a.....b.....c.....x
%C 1.....0.....1.....A124597
%C 1.....0.....2.....A198414
%C 1.....0.....3.....A198415
%C 1.....0.....4.....A198416
%C 1.....1.....2.....A198417
%C 1.....1.....3.....A198418
%C 1.....1.....4.....A198419
%C 1.....2.....1.....A198424
%C 1.....2.....3.....A198425
%C 1.....2.....4.....A198426
%C 1....-1.....1.....A198420
%C 1....-1.....1.....A198420
%C 1....-1.....2.....A198421
%C 1....-1.....3.....A198422
%C 1....-2.....1.....A198427
%C 1....-2.....2.....A198428
%C 1....-2.....3.....A198429
%C 1....-2.....4.....A198430
%C 1....-3.....1.....A198431
%C 1....-3.....2.....A198432
%C 1....-3.....3.....A198433
%C 1....-3.....4.....A198488
%C 1....-4.....1.....A198489
%C 1....-4.....2.....A198490
%C 1....-4.....3.....A198491
%C 1....-4.....4.....A198492
%C 2.....0.....1.....A198583
%C 2.....0.....3.....A198605
%C 2.....1.....2.....A198493
%C 2.....1.....3.....A198494
%C 2.....1.....4.....A198495
%C 2.....2.....1.....A198496
%C 2.....2.....3.....A198497
%C 2.....3.....1.....A198608
%C 2.....3.....2.....A198609
%C 2.....3.....4.....A198610
%C 2.....4.....1.....A198611
%C 2.....4.....3.....A198612
%C 2....-1.....1.....A198546
%C 2....-1.....2.....A198547
%C 2....-1.....3.....A198548
%C 2....-1.....4.....A198549
%C 2....-2.....3.....A198559
%C 2....-3.....1.....A198566
%C 2....-3.....2.....A198567
%C 2....-3.....3.....A198568
%C 2....-3.....4.....A198569
%C 2....-4.....1.....A198577
%C 2....-4.....3.....A198578
%C 3.....0.....1.....A198501
%C 3.....0.....2.....A198502
%C 3.....1.....2.....A198498
%C 3.....1.....3.....A198499
%C 3.....1.....4.....A198500
%C 3.....2.....1.....A198613
%C 3.....2.....3.....A198614
%C 3.....2.....4.....A198615
%C 3.....3.....1.....A198616
%C 3.....3.....2.....A198617
%C 3.....3.....4.....A198618
%C 3.....4.....1.....A198606
%C 3.....4.....2.....A198607
%C 3.....4.....3.....A198619
%C 3....-1.....1.....A198550
%C 3....-1.....2.....A198551
%C 3....-1.....3.....A198552
%C 3....-1.....4.....A198553
%C 3....-2.....1.....A198560
%C 3....-2.....2.....A198561
%C 3....-2.....3.....A198562
%C 3....-2.....4.....A198563
%C 3....-3.....1.....A198570
%C 3....-3.....2.....A198571
%C 3....-3.....4.....A198572
%C 3....-4.....1.....A198579
%C 3....-4.....2.....A198580
%C 3....-4.....3.....A198581
%C 3....-4.....4.....A198582
%C 4.....0.....1.....A198503
%C 4.....0.....3.....A198504
%C 4.....1.....2.....A198505
%C 4.....1.....3.....A198506
%C 4.....1.....4.....A198507
%C 4.....2.....1.....A198539
%C 4.....2.....3.....A198540
%C 4.....3.....1.....A198541
%C 4.....3.....2.....A198542
%C 4.....3.....4.....A198543
%C 4.....4.....1.....A198544
%C 4.....4.....3.....A198545
%C 4....-1.....1.....A198554
%C 4....-1.....2.....A198555
%C 4....-1.....3.....A198556
%C 4....-1.....4.....A198557
%C 4....-1.....1.....A198554
%C 4....-2.....1.....A198564
%C 4....-2.....3.....A198565
%C 4....-3.....1.....A198573
%C 4....-3.....2.....A198574
%C 4....-3.....3.....A198575
%C 4....-3.....4.....A198576
%C 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 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.
%e 1.4044148240924343641483279437457586037...
%t (* Program 1: A198414 *)
%t a = 1; b = 0; c = 2;
%t f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x]
%t Plot[{f[x], g[x]}, {x, -1, 2}]
%t r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.41}, WorkingPrecision -> 110]
%t RealDigits[r] (* A198414 *)
%t (* Program 2: an implicit surface of x^2+u*x=v*sin(x) *)
%t f[{x_, u_, v_}] := x^2 + u*x - v*Sin[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .01, 6}]}, {u, .1, 100}, {v, u, 100}];
%t ListPlot3D[Flatten[t, 1]]
%Y Cf. A197737.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 24 2011
%E Edited by _Georg Fischer_, Aug 01 2021