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Decimal expansion of x > 0 satisfying x^2 = 2*sin(x).
107

%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