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