A198755 - cp's OEIS frontend

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

Offset: 1

Clark Kimberling, Oct 30 2011

For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c..... x
... 1.... 2..... A198755
... 1.... 3..... A198756
... 1.... 4..... A198757
... 2.... 3..... A198758
... 2.... 4..... A198811
... 3.... 3..... A198812
... 3.... 4..... A198813
... 4.... 3..... A198814
... 4.... 4..... A198815
... 1.... 0..... A125578
.. -1.... 1..... A198816
.. -1.... 2..... A198817
.. -1.... 3..... A198818
.. -1.... 4..... A198819
.. -2.... 1..... A198821
.. -2.... 2..... A198822
.. -2.... 3..... A198823
.. -2.... 4..... A198824
.. -2... -1..... A198825
.. -3.... 0..... A197807
.. -3.... 1..... A198826
.. -3.... 2..... A198828
.. -3.... 3..... A198829
.. -3.... 4..... A198830
.. -3... -1..... A198835
.. -3... -2..... A198836
.. -4.... 0..... A197808
.. -4.... 1..... A198838
.. -4.... 2..... A198839
.. -4.... 3..... A198840
.. -4.... 4..... A198841
.. -4... -1..... A198842
.. -4... -2..... A198843
.. -4... -3..... A198844
... 0.... 1..... A010503
... 0.... 3..... A115754
... 1.... 2..... A198820
... 1.... 3..... A198827
... 1.... 4..... A198837
... 2.... 3..... A198869
... 3.... 4..... A198870
.. -1.... 1..... A198871
.. -1.... 2..... A198872
.. -1.... 3..... A198873
.. -1.... 4..... A198874
.. -2... -1..... A198875
.. -2.... 3..... A198876
.. -3... -2..... A198877
.. -3... -1..... A198878
.. -3.... 1..... A198879
.. -3.... 2..... A198880
.. -3.... 3..... A198881
.. -3.... 4..... A198882
.. -4... -3..... A198883
.. -4... -1..... A198884
.. -4.... 1..... A198885
.. -4.... 3..... A198886
... 0.... 1..... A020760
... 1.... 2..... A198868
... 1.... 3..... A198917
... 1.... 4..... A198918
... 2.... 3..... A198919
... 2.... 4..... A198920
... 3.... 4..... A198921
.. -1.... 1..... A198922
.. -1.... 2..... A198924
.. -1.... 3..... A198925
.. -1.... 4..... A198926
.. -2... -1..... A198927
.. -2.... 1..... A198928
.. -2.... 2..... A198929
.. -2.... 3..... A198930
.. -2.... 4..... A198931
.. -3... -1..... A198932
.. -3.... 1..... A198933
.. -3.... 2..... A198934
.. -3.... 4..... A198935
.. -4... -3..... A198936
.. -4... -2..... A198937
.. -4... -1..... A198938
.. -4.... 1..... A198939
.. -4.... 2..... A198940
.. -4.... 3..... A198941
.. -4.... 4..... A198942
... 1.... 2..... A198923
... 1.... 3..... A198983
... 1.... 4..... A198984
... 2.... 3..... A198985
... 3.... 4..... A198986
.. -1.... 1..... A198987
.. -1.... 2..... A198988
.. -1.... 3..... A198989
.. -1.... 4..... A198990
.. -2... -1..... A198991
.. -2.... 1..... A198992
.. -2... -3..... A198993
.. -3... -2..... A198994
.. -3... -1..... A198995
.. -2.... 1..... A198996
.. -3.... 2..... A198997
.. -3.... 3..... A198998
.. -3.... 4..... A198999
.. -4... -3..... A199000
.. -4... -1..... A199001
.. -4.... 1..... A199002
.. -4.... 3..... A199003
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 A198755, take f(x,u,v)=x^2+u*cos(x)-v 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.

			1.32562251814753662348322902938798744330...

Index entries for transcendental numbers.

Cf. A197737, A198414.

(* Program 1:  A198655 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.32, 1.33}, WorkingPrecision -> 110]
RealDigits[r] (* A198755 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A198755 *)

cp's OEIS Frontend

A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

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