A199863 -id:A199863 - cp's OEIS frontend

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

Offset: 1

Clark Kimberling, Nov 08 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 2.... A199597
... 1.... 3.... A199598
... 1.... 4.... A199599
... 2.... 1.... A199600
... 2.... 3.... A199601
... 2.... 4.... A199602
... 3.... 0.... A199603, A199604
... 3.... 1.... A199605, A199606
... 3.... 2.... A199607, A199608
... 3.... 3.... A199609, A199610
... 4.... 0.... A199611, A199612
... 4.... 1.... A199613, A199614
... 4.... 2.... A199615, A199616
... 4.... 3.... A199617, A199618
... 4.... 4.... A199619, A199620
... 1.... 0.... A199621
... 1.... 2.... A199622
... 1.... 3.... A199623
... 1.... 4.... A199624
... 2.... 1.... A199625
... 2.... 3.... A199661
... 1.... 0.... A199662
... 1.... 2.... A199663
... 1.... 3.... A199664
... 1.... 4.... A199665
... 2.... 0.... A199666
... 2.... 1.... A199667
... 2.... 3.... A199668
... 2.... 4.... A199669
.. -1.... 0.... A003957
.. -1.... 1.... A199722
.. -1.... 2.... A199721
.. -1.... 3.... A199720
.. -1.... 4.... A199719
.. -2.... 1.... A199726
.. -2.... 2.... A199725
.. -2.... 3.... A199724
.. -2.... 4.... A199723
.. -3.... 1.... A199730
.. -3.... 2.... A199729
.. -3.... 3.... A199728
.. -3.... 4.... A199727
.. -4.... 1.... A199737. A199738
.. -4.... 2.... A199735, A199736
.. -4.... 3.... A199733, A199734
.. -4.... 4.... A199731. A199732
.. -1.... 1.... A199742
.. -1.... 2.... A199741
.. -1.... 3.... A199740
.. -1.... 4.... A199739
.. -2.... 1.... A199776
.. -2.... 3.... A199775
.. -3.... 1.... A199780
.. -3.... 2.... A199779
.. -3.... 3.... A199778
.. -3.... 4.... A199777
.. -4.... 1.... A199782
.. -4.... 3.... A199781
.. -4.... 1.... A199786
.. -4.... 2.... A199785
.. -4.... 3.... A199784
.. -4.... 4.... A199783
.. -3.... 1.... A199789
.. -3.... 2.... A199788
.. -3.... 4.... A199787
.. -2.... 1.... A199793
.. -2.... 2.... A199792
.. -2.... 3.... A199791
.. -2.... 4.... A199790
.. -1.... 1.... A199797
.. -1.... 2.... A199796
.. -1.... 3.... A199795
.. -1.... 4.... A199794
.. -4.... 1.... A199873
.. -4.... 3.... A199872
.. -3.... 1.... A199871
.. -3.... 2.... A199870
.. -3.... 3.... A199869
.. -3.... 4.... A199868
.. -2.... 1.... A199867
.. -2.... 3.... A199866
.. -1.... 1.... A199865
.. -1.... 2.... A199864
.. -1.... 3.... A199863
.. -1.... 4.... A199862
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 A199597, take f(x,u,v)=x^2+u*x*cos(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.

			1.1881851344514388032178109729076525973...

Index entries for transcendental numbers.

Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429.

(* Program 1:  A199597 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]
RealDigits[r]  (* A199597 *)
(* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A199597 *)

Edited by Georg Fischer, Aug 03 2021

cp's OEIS Frontend

A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions