A199429 - cp's OEIS frontend

6, 4, 3, 4, 3, 6, 3, 6, 4, 1, 3, 8, 0, 2, 6, 1, 5, 8, 6, 4, 2, 0, 9, 8, 9, 1, 4, 3, 0, 4, 0, 1, 3, 1, 8, 2, 6, 8, 7, 4, 4, 6, 7, 2, 4, 1, 9, 4, 5, 7, 8, 5, 1, 6, 3, 2, 3, 8, 7, 4, 9, 1, 9, 8, 5, 8, 8, 7, 5, 2, 2, 9, 2, 2, 2, 7, 2, 5, 9, 4, 1, 7, 6, 4, 1, 7, 8, 8, 8, 7, 0, 7, 8, 5, 2, 7, 8, 5, 7

Offset: 0

Clark Kimberling, Nov 06 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c*cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 1.... A199429
... 1.... 2.... A199430
... 1.... 3.... A199431
... 2.... 1.... A199432
... 2.... 2.... A199433
... 2.... 3.... A199434
... 3.... 1.... A199435
... 3.... 2.... A199436
... 3.... 3.... A199437
... 1.... 1.... A199438
... 1.... 2.... A199439
... 1.... 3.... A199440
... 2.... 1.... A199441
... 2.... 3.... A199442
... 3.... 1.... A199443
... 3.... 2.... A199444
... 3.... 3.... A199445
... 1.... 1.... A199446
... 1.... 2.... A199447
... 1.... 3.... A199448
... 2.... 1.... A199449
... 2.... 2.... A199450
... 2.... 3.... A199451
... 3.... 1.... A199452
... 3.... 2.... A199453
.. -1.... 1.... A199454
.. -1.... 2.... A199455
.. -1.... 3.... A199456
.. -2... -3.... A199457
.. -2... -2.... A199458
.. -2... -1.... A199459
.. -2...  0.... A199460
.. -2...  1.... A199461
.. -2...  2.... A199462
.. -2...  3.... A199463
.. -3... -3.... A199464
.. -3... -2.... A199465
.. -3... -1.... A199466
.. -3...  0.... A199467
.. -3...  1.... A199468
.. -3...  2.... A199469
.. -3...  3.... A199470
.. -1...  1.... A199471
.. -1...  2.... A199472
.. -1...  3.... A199473
.. -2...  1.... A199503
.. -2...  3.... A199504
.. -1...  1.... A199505
.. -1...  2.... A199506
.. -1...  3.... A199507
.. -2...  1.... A199508
.. -2...  2.... A199509
.. -2...  3.... A199510
.. -3...  1.... A199511
.. -3...  2.... A199513
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 A199429, take f(x,u,v)=x^2+u*x*sin(x)-v*cos(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.

			0.6434363641380261586420989143040131826874...

Index entries for transcendental numbers.

Cf. A199370, A199170, A198866, A198755, A198414, A197737.

(* Program 1:  A199429 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .64, .65}, WorkingPrecision -> 110]
RealDigits[r]  (* A199429 *)
(* Program 2: implicit surface: x^2+u*x*sin(x)=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 10}, {v, u, 100}];
ListPlot3D[Flatten[t, 1]]  (* for A199429 *)

g(a,b,c)=solve(x=0,abs(a)+abs(b)+abs(c), my(S=sin(x),C=sqrt(1-s^2)); a*x^2+b*x*S-c*C)
g(1,1,1) \\ Charles R Greathouse IV, Feb 07 2025

cp's OEIS Frontend

A199429 Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).

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