A198414 - cp's OEIS frontend

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, 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, 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

Offset: 1

Clark Kimberling, Oct 24 2011

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 bc, they meet in quadrant 2.
Guide to related sequences (with graphs included in Mathematica programs):
a.....b.....c.....x
....0.....1.....A124597
....0.....2.....A198414
....0.....3.....A198415
....0.....4.....A198416
....1.....2.....A198417
....1.....3.....A198418
....1.....4.....A198419
....2.....1.....A198424
....2.....3.....A198425
....2.....4.....A198426
...-1.....1.....A198420
...-1.....1.....A198420
...-1.....2.....A198421
...-1.....3.....A198422
...-2.....1.....A198427
...-2.....2.....A198428
...-2.....3.....A198429
...-2.....4.....A198430
...-3.....1.....A198431
...-3.....2.....A198432
...-3.....3.....A198433
...-3.....4.....A198488
...-4.....1.....A198489
...-4.....2.....A198490
...-4.....3.....A198491
...-4.....4.....A198492
....0.....1.....A198583
....0.....3.....A198605
....1.....2.....A198493
....1.....3.....A198494
....1.....4.....A198495
....2.....1.....A198496
....2.....3.....A198497
....3.....1.....A198608
....3.....2.....A198609
....3.....4.....A198610
....4.....1.....A198611
....4.....3.....A198612
...-1.....1.....A198546
...-1.....2.....A198547
...-1.....3.....A198548
...-1.....4.....A198549
...-2.....3.....A198559
...-3.....1.....A198566
...-3.....2.....A198567
...-3.....3.....A198568
...-3.....4.....A198569
...-4.....1.....A198577
...-4.....3.....A198578
....0.....1.....A198501
....0.....2.....A198502
....1.....2.....A198498
....1.....3.....A198499
....1.....4.....A198500
....2.....1.....A198613
....2.....3.....A198614
....2.....4.....A198615
....3.....1.....A198616
....3.....2.....A198617
....3.....4.....A198618
....4.....1.....A198606
....4.....2.....A198607
....4.....3.....A198619
...-1.....1.....A198550
...-1.....2.....A198551
...-1.....3.....A198552
...-1.....4.....A198553
...-2.....1.....A198560
...-2.....2.....A198561
...-2.....3.....A198562
...-2.....4.....A198563
...-3.....1.....A198570
...-3.....2.....A198571
...-3.....4.....A198572
...-4.....1.....A198579
...-4.....2.....A198580
...-4.....3.....A198581
...-4.....4.....A198582
....0.....1.....A198503
....0.....3.....A198504
....1.....2.....A198505
....1.....3.....A198506
....1.....4.....A198507
....2.....1.....A198539
....2.....3.....A198540
....3.....1.....A198541
....3.....2.....A198542
....3.....4.....A198543
....4.....1.....A198544
....4.....3.....A198545
...-1.....1.....A198554
...-1.....2.....A198555
...-1.....3.....A198556
...-1.....4.....A198557
...-1.....1.....A198554
...-2.....1.....A198564
...-2.....3.....A198565
...-3.....1.....A198573
...-3.....2.....A198574
...-3.....3.....A198575
...-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.

			1.4044148240924343641483279437457586037...

Index entries for transcendental numbers.

Cf. A197737.

(* 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]]

Edited by Georg Fischer, Aug 01 2021

cp's OEIS Frontend

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

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